My wife and I went to Chenonceau twice, just over a week ago, on successive days. Both using our own car because we were staying nearby.
The first day we followed our hostesses' advice at our chateau and went at lunch time. She suggested that most people, and tour groups, would have stopped for lunch somewhere on the grounds and we would have a clear shot. Well, we just didn't see many people. It was Sunday. We actually found plenty of parking in the first row nearest the entrance.
We returned the next day, the Monday. That day we said we would just go after we finished breakfast and we got there about 10:30. Again parking was easy, 2nd row about 15 cars in. We would have been closer but there had been heavy rain overnight and there were puddles we wanted to avoid. At no time that whole day did we feel overly crowded. Oh wait, my wife felt crowded in the underwater kitchen, but it is a very cramped place. Anytime we wanted to sit down it was easy.
I'm sure that others have felt overcrowded but we honestly didn't.
There is a nonstop TGV leaving Paris Gare Montparnasse at 8:46, arriving at St-Pierre-des-Corps at 9:45, 20 minutes to change platforms and a train to Chenonceaux Chisseau at 10:05 arriving at 10:25 in 2 stops.