Door Buddy Cat Door Latch & Stopper. Adjustable Dog Proof Door Strap for Cats. Dog Proof Litter Box & Cat Feeder. Simpler Pet Gate Alternative. No Cut Cat Door Interior Door.

if you have room for a baby gate, they make them with a little door so that they're not so intrusive for adults- the point is that they are in the way, but not that they're making your life hell. due to the way my door is configured, i have a door buddy instead. it keeps my bathroom door just open enough for my cat to get in without letting my dog have a free-for-all in there. highly recommend. Door Buddy

I have a balcony similar to the pic you posted. I just got this door latch so I can keep the door slightly open for air without letting my cat out. It’s adjustable, so I’m planning to make the opening smaller than my cat can squeeze through. It’s a simple hook and eye type of thing so not obtrusive. Maybe your parents would be open to that? Also, not sure if you’ve had a balcony before or where you’re located, but they may end up not keeping the door open as much as they think, due to street noise, bugs, and climate control. Edit: the latch is assuming you have a hinged door. Sliding door they can easily keep open at a width smaller than your cat.