I assume you mean the fight against tamer [link eaten by forum], who actually uses 2 undead ([link eaten by forum], [link eaten by forum])and 1 dragonkin ([link eaten by forum]). To look for attacks that are super effective against those pets, you'll need attacks of the Critter family (against undead) or humanoid family (against dragonkin).
The "Abilities OF [insert pet family here]" actually allows you to find those abilities (the ability family does not need to be the same as the family of the pet using that ability). You'd then need to click on all the abilities, find which pets use them and see if you can capture those. More convenient is to use the Pet Battle Arena: [link eaten by forum] and use the advanced search options, selecting for example "strong against Dragonkin", "caught in the world" and "level 14-25". Note this results in all pets that have a single ability in that family, for example rabbits with [link eaten by forum] will show up while that attack doesn't even do damage.
Besides looking at dealing high damage to the opponent pets (by picking correct ability families), you can also try to look for pets in families that take reduced damage from the tamers pets and want to avoid pets that take increased damage from their attacks. In this case the tamers' pets deal undead, dragonkin and elemental damage, so look for an aquatic, undead or critter pet and avoid humanoid, magic or mechanical pets (or swap them out when they are against a strong opponent).
Critters with critter attacks are easy to find (many even have [link eaten by forum] to help against the dragonkin, albeit with a cooldown). Aquatics with critter attacks are easy to find in frogs or turtles. Humanoid attacks are rather rare or on cooldown. Flayer Youngling or Sporeling Sprout come to mind to help fight against the dragonkin (since he's not the one with undead damage).