I want to sign up for skating lessons as a 16 year old? Do I need parents permission or can I just call the rink and sign up myself?

No you cannot sign yourself up. With registration there should be a part to sign as a waiver. That waiver is saying that if you get hurt you can’t sue and that by signing you’re acknowledging that rule. That needs to be signed by an adult

And don’t listen to that idiot Chris A, you CAN NOT teach yourself how to skate properly

You do need permission. Also, calling the rink will not do, you have to physically go there. When arrived, you talk to the manager and schedule all the payments and rents. They have a sheet where you OR YOUR LEGAL GUARDIAN THAT IS OVER 18 have to sign. And, Chris Ancor, I am a professional skater, but your story is different. There are 3 types of skating. One is hockey, one is speed skating, and one is figure skating. Yes, you can learn by yourself for speed skating, but hockey and figure skating take years and years to master. 

yes you do need permission. but after all the application signing, you will be able to go to the lessons by yourself since you are over 8

No you don't, but then again lessons are not needed initially. You can teach yourself.

No idea. Ask the skating rink.

Nobody calls places, troll. Sign up online