How about a page that lists all of your pets in easy-to-sort columns? Kind of like a spreadsheet. 

...here's a Football Manager screenshot to illustrate this suggestion:

Basically, a list with sortable columns. 
It could have columns for pet name (with a link to profile), energy, species, gender, level, color, stats, dominant genes, and so on. Maybe with some icons to indicate adoption, childhood, pregnancy, genetic power (like in the arena), etc. 

That'd make it a lot easier to find and sort pets in a large stable, and you could fe. tell by a glance if you have any pets whose stats are close to what's needed for the daily quests..