Frankly you and your parents are both being very very weird about this.
An airbed or cot can be purchased on Amazon for under fifty dollars. Respect other people's personal space. Adult humans are unaccustomed to being physically close unless they expect sex, and families have a *lot* more sex with each other than you think. Neurotypicals basically have to have sex if they so much as brush past each other in a hallway. You should know this from observing their behavior. Fifty dollars. Airbed. Buy one.
If your cousin is the one pressuring you to share the bed with her, just make sure you use a condom for God's sake. Don't make any web-toed babies.
Edit to add:
And I don't think anybody thinks you might assault her, it's just that physical closeness makes people act irrationally and inappropriately due to the neurotoxins like oxytocin that are released by the body. Basically, touching leads to sex whether they intend for it or not. And your Asperger's doesn't make you immune to being affected by those neurotoxins, you just might perceive them differently. Even when nobody is being assaulted, humans "accidentally" have sex with people they didn't intend to, all the time.
Once the neurotoxin oxytocin is released, they are pretty much going to try to "accidentally" have sex with the nearest living thing. It's kind of gross, really. Also, if you and cousin didn't grow up together, the sexual response to physical closeness could be involuntary for both of you. You wouldn't have the instinctive boundary preventing sexual attraction that occurs with a close sibling, or someone you grew up with.