tough situation with ex bf.

Alex • 25, ABD&OB/GYN RDMS
Me and my bf have been off and on for 4 years... broke up 3 times. 
He broke up with me in late October because he felt that he couldn't handle a relationship at the moment and also because I have jealousy issues. He came back right after Valentine's Day asking for me back saying he made a mistake and wants me back and for good. I finally met him up to see how sincere he was or wasn't. His voice quivered and he assured me he knew what he wants now and said that he should have just stuck by my side through the jealousy problems. I told him I wasn't a changed woman so if he was expecting that then don't ask for me back. He said he wasn't. I brought up the fact that my parents highly dislike him because of the back and forth getting back together and breaking up. I was like you're not scared of my parents? He said oh yes I'm scared of them but that's how much I want you back and I'm willing to do whatever for you and for them to trust me again. He was saying this time it's for the long run and that he wants to ask my hand in marriage in the next couple years. He knows I'm skeptical. I have a lot of people that I work with support and my older sister and some extended family. As far as my parents know they just know he's been texting me begging and asking for me back. I didn't tell them I'm hanging out with him again. I've been keeping us really private from my parents. I told him we have to private for a long while. I feel like if they knew he gave me more time they may be alittle more okay with it and feel more like he respected my time. I'm really really scared. Of corse I want him back he's like the guy I'm supposed to marry. We had always talked about it atleast. My moms reaction to me telling her about him texting me was like "Oh I knew this was coming... you better not get back with him. I know I can't tell you what to do but I pray you don't go down that road again."
My dad was pretty just like whatever about it and joked about telling him to "go find a nice Vietnamese woman" (he's Vietnamese). Please y'all tell me your opinions. I'm scared to death. I feel extra compelled to tell them now that I want him back and get it overwith. But should I just give it more time and act like he's been begging for a long time?