OP / their daughter can switch to an F1 student visa and stay on. But after graduation they’ll need to find a sponsoring employer to switch to a work visa.
Unfortunately yes, that’s a big problem with our immigration system. I have several employees in a similar situation. Since OP is from India, wait times for employment based green cards are exceeding 13 years. Unless OP / parents are very close to obtaining their permanent residency, an independent work sponsorship will be required after graduation.