@highsckid - Please tell your parents that I know a student who scored lower than 800 on Math Level 2, was accepted by Princeton, and went on to graduate magna cum laude in Physics.
@theloniusmonk first of all, mastery of a topic isn’t dependent upon your race. Second, I agree with you that at the very highest level, the SAT test doesn’t differentiate the top. That is why if you look at the Harvard classifications, to get to the highest academic one you have to have something outside of normal standardized tests. They would lump in a 780 with an 800. Maybe if I used the world competency you would like it better. At the level that the SAT2 can measure, a 780 and an 800 both show that you have a base competency in Math. If you want to try to show more, you have to do something else (e.g AIME).
Is a 780 a good score, yes. Do most students applying to t20s have better scores, also yes. I think it would be worth to retake because you have time and schools like MIT/others have no way of knowing it is the third attempt (if they require all scores that is just for SAT 1 and ACT, not SAT 2). I took it twice and got 780 both times and then gave up (arrgh). The extra 20 points will barely improve your application but keep in mind that the ivies are high reaches for those with perfects. There are countless good engineering schools that will take you for your 780 so build a list of safeties and matches. Your mother is partially correct in that engineering schools have a much higher standard in math but don’t require a perfect. They would rather see a well-rounded applicant with good ECs and essays. She is absolutely incorrect when she says you will fail multivar. I got a 780 and BC calc is one of my easiest classes (BC is 10000000000x harder than multivar at my school). Your parents are well meaning but they need to understand that an 800 won’t improve your chances at a t10.
TLDR: Only retake math 2 if you have the time and already have your desired SAT 1 score (much more important than the subject tests). The only colleges this will improve your chances at are t10s who are high reaches for every person on earth regardless of qualifications.