"On the morning of Christmas Day" is correct, though it is more natural to say, "On Christmas day morning". "On Christmas day morning, I visited my parents".

"on" is correct. Saying "on the morning of Christmas Day" would be a way to describe December 25th in a situation where Christmas wasn't being celebrated. For example, "On the morning of Christmas Day, their ship crossed the international date line 500 miles from Midway Island." "Christmas morning", as suggested by Hilary Metcalf, is more common. It evokes the image of some kind of Christmas celebration.

Thank you for your help.

I would go with Hilary's version.

you can also say "On Christmas morning" - you don't need to add the word 'day'. On Christmas morning we opened our presents.