This was my thought. If OP wants to keep helping her parents she could keep the account small and have that be an allowance for her parents. But in no way should they have access to any money she isn't willing to lose.

Them taking money before asking is definitely an issue. Even if she was going to say yes they're too comfortable with just taking money from her. If they aren't poor but aren't well off, they can probably work on their own budgeting to not need the allowance. They're so used to OP providing for them that they're dependent on her for the extra income, and the guilt tripping makes it even more of a requirement to restrict their access.

Since this is your first apartment, I would HIGHLY recommend learning about your local tenants' rights. Many landlords are either themselves unaware or count on renters not knowing the details, but some places have numerous restrictions on how the landlord has to handle security deposits and last month rent prepayments.

Plus knowing your rights regarding privacy and landlord access and all that will help you to avoid some scummy practices.

It's pretty standard in MA, though often it's more like 3 of 4 for first/last/security/broker fee.

In MA, at least, they technically need to put anything prepaid (security and last month rent) into a savings account and give you the interest. Or a flat 5%/yr if they don't put it in a bank account. I can't say I've really seen the interest but usually it's not enough to bother pushing as long as they give me my full deposit back. Many might not even know they owe this, though some probably just count on tenants not being aware.