Do I need to file taxes?

I’m married filing separate, however, I have been getting financial support from my parents since I am not legally separated. Can they amend their taxes to claim me?

 

I didn’t earn any income from a job or my business, because I am the sole caretaker for my small child. Do I need to file my taxes with no income to report? I didn’t file for an extension, is it still possible? How can I get a Schedule C if I didn’t file because there was no income to report?