Given that you appear to have difficulty articulating your thoughts in writing, you should consider having somebody help you compose your posts so that they're understandable to other people.

If somebody has a restraining order prohibiting you from being on the premises, you are supposed to obey that restraining order.

If you have been banned from the apartment building due to your past conduct, you are supposed to respect that ban. While there could be narrow exceptions that would apply to a social visit, depending on the full facts (and in the absence of a restraining order), a tenant would put herself at risk of eviction if she knowingly allowed you to move into her unit despite the ban, and may face consequences for allowing you to move into the unit without reporting the additional occupant to the landlord. Your parents will have to review their lease to see what it provides -- but, from the rest of your post, it doesn't sound like any of that would be relevant.

If your parents were given a 60-day notice, that suggests that they are month-to-month tenants and that their tenancy is being ended. When you rent on a month-to-month basis, that's one of the risks you accept. A landlord does not need special grounds to end a month-to-month tenancy upon proper notice. Yes, your parents owe rent through the end of their tenancy.