Good morning, and thank you for your post:
"I would like to apply for Social Security disability and I have two children in my care. I wanted to know what my rights are as a parent receiving benefits while caring for my children. Depression and anxiety run in my family and I have not been able to work in over a year. I have not yet applied but plan on doing so in the near future."
Your rights are not different than any other parent or non-parent to file a claim for Social Security Disability. Being able to caretake for your children, by itself, does not typically mean you can work a 40 hour work week in a simple but competitive work environment, and that is the general standard of whether we are disabled as that term is defined by the SSA.
So, again, you gain no additional rights just because you are a parent, in your claim for insurance benefits, but you are also not deprived of any rights simply because you are a parent.
Good luck with your mental health struggles and your claim.