How do I unlock the next level on Pet Rescue Saga without asking friends?

Unless you buy tickets to pass the next level, I think your only other option is to ask friends. Why don't you check out the Pet Rescue Saga Facebook page, here you will be sure to find loads of players who love the game I am sure that they will help you too. 

I suggest you join a facebook group to find more friends ho also play the game - you can use your facebook settings to restrict 'game friends' to only see your public profile and keep your personal stuff private. I can highly recommend the group at this link (copy and paste it into your browser) https://www.facebook.com/groups/1508327682775134/