I have Oak leaf Hydrangeas in the front yard that has done well with morning sun, It doesn't like the hot sun. I now have about a doz new one coming up under the big one. They do well even with a hard long winters here in New England
You have one of the smartest parrot around. I have a 35+ year old Red Lored Amazon who has a nasty mouth when she talks she gets her Ts and Fs mixed up and has leaned other words she doesn't need to use. But I'm lucky she doesn't talk as much any more. I still lover her even when she bites me.
I've been retired 10 years now and my neighbors still think I'm crazy for spending so much time in the garden. I grow only veggies that I know I'll eat and plants things I like. I also have 5 aquariums that keep me busy all winter.
I would just cut it as low as you can and keep it short and most of the vine should die out. My back lawn is full of clover and I enjoy watching the bees and bunnies play in the clover. and if the dog digs in the lawn it won't hurt it.