Chrom: Chrom will increase Inigo's offensive abilities through his Strength, Speed, and Skill, but still leaves his defense growths a little low. As the son of Chrom, Inigo will always inherit Rightful King, which will increase the activation rate of skills, though mostly is used for his offensive skills and the Aegis skill through his inheritance from Chrom. He does not inherit the Lord class, but he does inherit the Cavalier and Archer Classes. Like Chrom, Inigo could perform well as a Sniper, but given that Sniper is limited to indirect combat, it's best to leave him in a more versatile class if seeking the maximum offensive output from him. Cavalier, on the other hand, is a much better option for offensive flexibility. It provides him with Luna which is a great offensive skill for its good activation rate and is even better on higher difficulties. Dual Guard+ is good to use to further prevent any damage to Inigo as long as he is in a Pair Up. Finally, as mentioned above, Aegis is good because it, as well as Luna, build nicely off of his Rightful King for very high activation rates (61% without Limit Breaker, 71% with, not including skill boosts). Alternatively, giving him Lethality can also work well, as when used with Limit Breaker, will give him a 1 in 4 chance of pulling off the skill, which can kill almost any unit instantly without much hassle.
The Avatar: Because the Avatar has many different Stat Asset and Flaw combinations, there is no clear way to gauge how Inigo will run stat growth and cap wise if he is the son of the Avatar. Nevertheless, Inigo becomes a solid unit as the Avatar's son thanks to his ability to reclass into every male class possible. Though his stats growth and max stat modifiers will be exactly the same as his sister, Morgan, Inigo will have access Male class options and their skills. Regardless, Inigo becomes an extremely flexible unit and can be reclassed into other classes to fill roles when needed or is perfectly fine building a skill set that suits the player's style if keeping him in his base class.
Frederick: Frederick unfortunately removes Inigo's speed inheritance from his mother, but in exchange, his skill receives a nice boost as well as some good patching to his Defenses. Inigo gains the Cavalier, Knight, and Wyvern Rider class sets. Like for a Chrom fathered Inigo, Inigo should grab Luna, Dual Guard+, and Aegis for good offense and defense. The Knight provides him with the General promotion which ultimately gives him Pavise to shave off direct weapon damage. The Wyvern Rider class provides him with some good skills to use like Swordbreaker and Lancebreaker to ensure that he becomes a good offensive unit that can ensure a hit and avoid getting hit.
Virion: Virion is a modestly good father for Inigo stat wise, giving him a great boost to his Speed and Skill and a little to his Strength. Inigo can spend time in Virion's reclassing options (Archer, Mage, and Wyvern Rider) to collect necessary skills. If considering to use Inigo as a Sniper, Virion's stat mods are the best suited for Inigo if this route is chosen. In many ways Inigo will outpace his father thanks to his wider skill set and his offensive skills. Mage, on the other hand, provides him with the Sage class, but his stat inheritance is not suited to pursue this role in comparison to a Ricken-Inigo. Dark Knight, on the other hand, gives Inigo his only regenerative skill, Lifetaker and with his high speed and skill, he can take down just about anyone. Wyvern Rider Virion-Inigo again benefits from his high speed and skill boost and its own high Strength cap means that Inigo can hit fast and hard. If choosing not to keep him in this class, Swordbreaker and Lancebreaker are great skills for Inigo to use.
Stahl: Stahl as a father is very similar to Chrom stat wise, boosting Inigo's Strength, Speed, and Skill. Stahl also gives a boost to his Defense but loses some Luck. All in all, Stahl is another balanced father for Inigo. Reclassing wise, Inigo's reclassing will work exactly the same as Chrom so consulting Chrom's section above for the skills to learn will be the same.
Vaike: Vaike increases Inigo's strength the most out of all of his potential fathers and gives him a solid boost to his speed and skill. Unfortunately he drops a bit of Resistance in exchange. Inigo gains the Fighter and the Thief class options. Inigo does not benefit from the Fighter inheritance since his Mercenary and Barbarian inheritance from Olivia already gives him the Fighter's promotions. Thief on the other hand provides him with the Trickster promotion, though the skills inherited from it Lucky Seven and Acrobat are very situational.
Kellam: Like Frederick, Kellam also removes Inigo's Speed inheritance from his mother. However, he offers solid Strength and Defense in exchange, and the Knight, Priest and Thief class sets. Knight's promotions offer Pavise from General which can weaken direct hits, and Luna from Great Knight to enable Inigo to hit harder. The Thief and Trickster class provide more situational skill. Priest offers Miracle, which could potentially save Inigo's life in a pinch. Priest offers the War Monk promotion, from which Inigo can pick up Renewal which is a strong regenerative skill.
Donnel: Unfortunately, for the most part Donnel is a lacklustre father for Inigo. While he does offer high Luck and reasonable Strength, Inigo gains no new promoted classes since he already has access to Hero and Warrior. Fighter offers the modestly useful HP +5 and Zeal; while Villager offers Aptitude which can help Inigo max his stats faster. Remaining in the Villager class is mostly a liability for Inigo due to its low stats, and that is the only class that Donnel-Inigo has that uses Lances. Unfortunately, this pairing gives Inigo no new promoted classes, so it should generally be avoided.
Lon'qu: Lon'qu gives Inigo the best Speed and Skill modifiers, and decent Attack, Magic and Luck ones too. Unfortunately, this Inigo won't be taking many hits due to poor Defense and Resistance modifiers. Fortunately though, the Wyvern Rider set Lon'qu passes somewhat alleviates this, as they give Swordbreaker and Lancebreaker. Inigo can make a good Griffon Rider, with inherited Galeforce, Axefaire and Swordbreaker. Be sure to keep him away from magic users though. Thief doesn't offer anything particularly useful, while Inigo's low Magic growths make Trickster unsuitable. However, having access to the Thief classes, as an Assassin he can learn Lethality (but still not as helpful without Rightful King from a Chrom-fathered Inigo), along with Pass, to avoid being cornered and Acrobat to move throughout all terrain without hassle.
Ricken: Ricken gives Inigo good magic, Speed, Skill and Luck, although slightly reduces his Defense. Ricken passes down the Mage, Archer and Cavalier class sets. From Sage, Ricken-Inigo can grab Tomefaire to bolster his Magic and from Dark Knight he can grab Lifetaker, which will allow him to have a healing skill that won't interfere with his pre existing Astra and his newly acquired Luna. Cavalier's promotions offer the ever useful Luna and Aegis, which Inigo should take the time to pick up at least one. Inigo can also be a good Sniper, although Chrom, Stahl and Virion are preferable fathers if you wish Inigo to be a Sniper since they pass better physical modifiers. Overall, this Inigo is best left as a Sage. Although his Magic growth is quite low, he will maximize at an impressive 49 Magic which can be bolstered to 54 with Tomefaire.
Gaius: Gaius offers Inigo good Speed and Skill modifiers and solid Strength, but unfortunately forces him to sacrifice some Defense and Resistance. In terms of new classes, Gaius offers the exact same classes as Vaike, so consulting Vaike's section above will give you a good idea of how he'll perform in those classes.
Gregor: While Gregor's modifiers are decent, with only Resistance being in the negative, unfortunately he gives Inigo no new classes whatsoever. While Inigo will still be good, in terms of both modifiers and extra classes there are much better fathers to give him.
Libra: Libra opens up the spellcaster path for Inigo, passing him the Mage set, the Dark Mage set and the Priest set. From Mage's promotions, Inigo can grab Tomefaire to help bolster his Magic and Lifetaker to replenish his health. Sorcerer offers an attacking skill in Vengeance, which Inigo can combine with his pre existing Vantage and Wrath to make him extremely dangerous at low health. Sorcerer also offers Tomebreaker, if you need an anti-Mage. Priest offers Miracle, which could save Inigo's life in a pinch while War Monk offers a strong regenerative skill in Renewal. Overall, Inigo will perform well in all of his new classes. With inherited Galeforce and his automatic Armsthrift he can run around with forged Celica's Gales to be a powerful offensive Sage. As a Sorcerer, he can either run a high risk high return build with Vantage, Vengeance and Wrath or a more defensive build with Sol, Armsthrift and forged Nosferatus/Aversa's Nights. He can use Dark Knight and War Monk well too, as he has the relevant Fares for both class sets.
Henry: Henry is functionally much the same as Libra, although he doesn't offer as much utility. If you wanted to make Inigo a Trickster, Henry is probably your best bet as he will have higher Strength and Magic than Lon'qu and better Speed than Kellam. As a Dark Kinght or Sorcerer, he will perform much the same with Henry as his father as he would with Libra, albeit slightly worse since this Inigo will lack Tomefaire.