Only issue is that if they need to lie about "you checking in later", it won't add up if the booked room only has one bed and two people who are not you turn up to check in.
I'd book the 2 twins anyway, since it's only base room available (they don't really care about upgrades or a bigger room). I guess they can just have my mom sit in the car, and say my dad is checking in for the both of us, and I can book a prepaid/non-refundable rate?