We were really smashed in for a while, too. I think, like the other moms have said, that you need to keep the 20 month old right there in the crib. Use white noise to muffle the sounds of New Baby crying so 20 month old can stay asleep. Earplugs and white noise will also help Dad sleep through and still remain in his own room.
(If you can't afford a white noise machine (we sure couldn't) get an atmospheric CD of rain or the ocean and play it at night to blend the sounds she hears. At one point, (I kid you not) I even tuned the radio to soft static. Worked fine. Even with 3 kids in one room, all under 4.)
Don't feel bad about your parents' sleeping arrangements. Remember they're here to help you. When they leave, they'll want to leave knowing that you have an arrangement and a routine with which all of you can be comfortable. That means supporting all that hard work you did getting your toddler into her crib.
PLUS, your toddler needs that security -- she's little, but she's not dumb; she'll put 2 and 2 together real fast if, the moment the new baby comes along, her whole life turns upside down. She needs to feel secure with her position in the family *and* in the house. That's her space. Let her keep it. (Translation: in my experience, Yup -- shifting her into a routine where she wakes up with a parent beside her will totally mess up all the independent sleeping progress. This is the road to long term co-sleeping, which, though I loved it, sure did keep me awake a lot.)
Let your folks do this for you. Get them as comfy as you can in the living room and remember that none of this is a mystery to them, they've been through it before.
And by the way -- having kids close together rocks. Absolutely the coolest, craziest, most magical challenge you'll have until... well, until you have grandkids and get to help *your* daughter out after *her* C section. Congratulations!!! To all of you!