I wouldn't have a problem with a parent not staying and you are probably an ok person, but if I felt like a parent was uncomfortable with me staying, especially if my child hadn't even been in their home and wanted me to stay, I would consider it a red flag and wonder if there was something to hide. Again, probably nothing to hide in most cases, but my parent radar would go up.
And in the case of my son, if he's more comfortable with me staying, why not? I have older children (oldest is in college) and I know he will be out of this stage soon without me rushing him, so why push him if I don't have to (ie, another engagement that I have to go to, etc).