How do I handle this tricky information exchange between my old and new employer?

Q: “I have a job offer and the employer wants to know the start date of my prior job. The problem is that my old job’s HR department determined my start date from the time I received benefits, while I was technically employed prior to receiving benefits as a trainee with a reduced salary. I sent my prospective employer an old email that documented the training start date from a former supervisor—is this sufficient to prove the earlier start date? Does my new employer have the right to contact my former supervisor regarding this matter without my permission? My old supervisor isn’t the easiest person to speak with, which is why I hope the email is fine.” – Fred, Massachusetts