Back to childhood rules, it's only funny if everyone is laughing.
I'd just respond to the "You're so sensitive" with something along the lines of "Well, now that you know I'm sensitive about it, cut the shit" and don't budge or apologize for any sulking if you get snappy. I've had to use that line a few times. You look like a kill joy for a short amount of time, but really if they respect you, they'll respect the boundaries you put up.
Have you told your DP how annoyed you are? If he doesn't take you seriously, you need to ditch him. Life is too short to be with someone who doesn't respect you.
I agree with PP's this is a form of bullying if they know how upset you are about it. If you're laughing or sort of going along with it, they may not know how serious it is.
From previous experience you have to nip shitty behavior in the bud or people will all happily walk all over you. The more you let this go on, the harder it will be to make it stop. I had a work bullying situation that I thought ignoring and carrying on would be the best course of action. When I tried to get HR involved to make it stop, it was already too far out of control. Though, I felt smugly vindicated the branch was shut down shortly after I quit due to shitty management. Fuckers weren't laughing so much then.
Anyway, don't play nice with this, and don't be afraid to be assertive or "mean" about it.