I agree with Mallory and Annette. For the first play date, it's perfectly reasonable to let the hosting parent know that your daughter isn't quite comfortable yet being dropped off alone for a play date at a house to which she hasn't been before. Let the other parent know this when your daughter is invited so that the play date can be scheduled for a day and time when it is convenient for the other mom to play hostess to both your daughter and you. While you're there, it's perfectly reasonable to ask about things like pets, weapons, pools, trampolines, etc. If your daughter gets comfortable and you get a good vibe from the home, you can try to leave after a half hour and see how your daughter reacts to something like "I'm going to run a few errands - you can stay here and I'll pick you up at X time" and if she balks, let her know that you'll both stay for a little while longer and then will go. If you need to stay the whole time, limit the play date to about an hour (unless things are going really well and you and the other mom have tons to talk about and time is flying).
As for subsequent play dates, I have to say that honestly, I wouldn't do too many play dates with a family in which the parent needed to stay the whole time for more than one or two visits. As Mallory said, I schedule play dates to occupy my child, not to play hostess to a mom who is not really a friend (yet). I work from home and really don't have an hour or two to carve out in the middle of my work day to chit-chat because someone is a nervous nellie. In that case, I would suggest that the nervous family host most of the time. Staying for the whole time once (maybe twice) or for a few minutes at the beginning of many play dates is totally reasonable. Expecting to stay for the whole time on a regular basis would be a total PITA.