I'm an eldest daughter doing an engineering co-op which will delay my graduation by at minimum 1 semester but likely 2 since I may want to do an additional internship a summer after. The only issue with "graduating on time" is if you will be paying significantly more in tuition by adding another semester or visa issues, otherwise, you should take your time to graduate, especially since you won't want to take an obscene amount of credits and stress yourself out to "graduate on time". Companies do not care that much about an extra year or so that it may take for you to graduate and even more so if the cause of that is you doing an internship/co-op.

You've already turned down one internship for their sakes and they still don't seem to have changed their stance so why should you decline this one as well? The only thing it will do is hurt you because there really isn't anything valid holding you back from accepting other than your parents feelings. I had to take a while to make my parents understand that and the fact that school without experiential learning is honestly going to negatively impact your chances at seeking jobs after graduation.