If I invite them over to play, what should I check with the parent?

It's reassuring to check in advance with the other parent about food allergies and dislikes (picky eating, sensory issues), things that cause trouble and ways to calm down, contact details, pick-up and drop-off, and needed accommodations. Asking without being too formal—with the attitude of "tell me so we can have a good time"—gets through better.