Odd Language #161: Birthday

Birthday wishes to a regular on a USENET newsgroup that I follow:
“Marking the 81st anniversary of your birth on July 9, 2016,
I wish you all the best. Live long and strong.”

The issue: Is the first line grouped “[Marking the 81st anniversary of your birth] on July 9, 2016,” or “Marking the 81st anniversary of [your birth on July 9, 2016],”? It is the former, but this line did trip me up briefly.