Pet Missions ask for a specific number of Pets. Rescue them within the normal Levels of the game.
You can just continue from the Level you are at, or go back in the game to re-play previous Levels.
The screen at the beginning of your game will show you what the reward for your mission is.
Any Pets saved will go toward the total, you do need to complete the level for the Pets rescued to count.
If you want to check your progress, just click on the little dog kennel in the corner of your game, it will tell you how many Pets you have already saved.
Here is a list of some levels that you can use to get a lot of Pets at a time.
(If you click on the links you will see an example of how to play each Level)
Level 14 (16 pets),
Level 15 (16 pets),
Level 16 (10 pets),
Level 21 (16 pets),
Level 96 (14 pets),
Level 218 (36 pets),
Level 315 (26 pets),
Level 705 (35 pets)
Level 834 (25 pets),
Level 934 (25 pets),
Level 983 (20 pets ),
Level 1023 (20 pets),
Level 1030 (30 pets),
Level 1071 (20 pets)
Level 2055 (62 Pets)
Level 2064 (32 Pets)