The first play through example in the video, shows how to get a few Pets down in the top section, then start clearing balloons to help them drop.
Use the rocket/s built up within the game to set off whole rows of balloons to remove as many coloured blocks at a time as you can..
The second play through concentrated more on the balloons, popping and clearing them in rows or by choosing colors to see if it would result in getting the Pets down more efficiently.
Both methods resulted in saved Pets, but getting the Pets first, then popping balloons resulted in a better score in this instance.