You discussed the 13th. You stated that would be possible. You asked for specific times.
What I would do now is either send them an email or however you wish to respond, and state specifically what will work for you that day.
Mom and Dad, as we discussed, you had stated coming down on the 13th to visit. Here is the time frame that will work for our schedule that day. Please confirm final details by Dec. 31st, so that we do not make other plans.
Details should be an approx. time they are coming. With a small window. Say, arrival between noon and one. If they do not arrive by one thirty, I would be out the door for a shopping excursion or something. And when they call and ask where you are, "Oh, so sorry. I thought we agreed upon such and such a time? You never arrived. So, I thought you had changed your plans. Sorry, but we cannot make it back in time now. We are at mall. You are welcome to come visit with us for a few minutes before you head back home or we can arrange another date and time".
It seems that you have to be specific about when mom and dad visit and the plans involved. Same if you go to visit them. And always be firm.