Your parents situation is pretty typical. Your mom has a much lower income than your father, so she will probably be better off taking the spousal benefit from social security. Women generally out live men, so maximizing her income should he die first is an important consideration. In all likelihood, your father should not file for social security now, but instead wait for full retirement age (FRA). She will get half his benefit which will be more than hers. There are social security planning programs, but your best bet is to read up on FRA and spousal benefits so you fully understand the choices. BTW, full retirement age for your parents is 66, not 65.
At 65, your parents will be eligible for Medicare. Until then, they will need health insurance. Does your mom have employer-provided health insurance? Is it possible to cover both of them through her employer? If not, did your father receive a severance package? Is COBRA available? If so, you will have to figure out whether to use the COBRA plan or move to a private/ACA plan.
Keeping your mom's salary is part of deciding where to live. If she is ready to retire and move south, that's great. If she wants to keep working and they want to stay on Long Island, her salary is an important piece of the retirement puzzle. Does she have a 401k or a pension in addition to her IRA? If she is adding to those, or she can cover both of them under her health insurance, working longer has important benefits that should be considered.
I could go on, but I think you get the idea that retirement is a complex set of decisions that will take time and research to work through. The most important thing is not to make rash decisions that cannot be undone.