I second arila's logistical concerns--just picturing the engagement-type parties I've been to, there's been no point in the event when all of the guests trail past the same point in a steady line, such that one could set up a receiving line. If there was such a point, it would probably be when everyone heads to the food table to serve themselves, and they naturally form a line. That seems like an odd time and place for a receiving line to be set up, but I guess there's no reason why not. Maybe the "receiving line people" could actually be the ones moving, walking down the line of people waiting to get food and shaking people's hands, being introduced, etc..
If they are "clenched teeth amicable," then I think they can stand/walk near each other well enough. They don't have to interact with each other if the line of guests is steady--they'll be focused on the person they're being introduced to. I think awkwardness would come in if there were frequent lulls in the line of guests, such that it would be natural to talk to the other people in the receiving line.
Perhaps rather than a line, you and each of your parents could take one "group" of future in-laws--your dad with your DF, your mom with DF's mom and boyfriend, you with DF's dad and partner maybe--and be formally responsible for introducing them to all the other guests by walking them around the room or catching guests near the door. Since the guests are your friends and family, you could prepare in advance something of interest to say about each one: "DFDad, Partner, this is my friend Sue. We were roommates in college and she's a big Yankees fan" (as is DFDad). They would exchange a couple of sentences, then you could move them on to the next guest. Then when you finally turn your future in-laws loose, they could mingle with the guests on their own, without having to start from a base of zero. Think of it more like when the HC walk around to each table of guests at their reception, instead of a receiving line.