Your parents are doing more damage to you than your boyfriend by threatening abandonment. How curious the very thing they say they don’t like that your boyfriend did — hurt you by pulling away — is what they are threatening to do permanently. Your boyfriend at least had the guts to recognize his mistakes. Your parents have created a lose-lose situation for you. If you break it off with your boyfriend because of them you will resent them for usurping your own decision-making power. If you stay with him you lose them. Parents who use this type of power play typically paint themselves into a corner because in either case they have a daughter that isn’t very happy around them.
I would sit down with your parents and outline what you need from them. Let them know this is your life and you want to make your own choices and mistakes. Explain that they have created a situation where everybody loses, and you would like a chance to see this through. Maybe your boyfriend isn’t right for you, but it will have to be your decision, not theirs that has to come to this point.
The goal of the discussion is to move from conflict to cooperation. Ask them for some time to assess the relationship. Ask for three months or whatever you feel is reasonable, to let you engage in this relationship, with the understanding that the three of you will sit down in three months to talk again. Move this conflict into dialogue and experimentation. I would love to hear from you down the road to see how this worked out.