Unfortunately, the job of Palace Guard now includes keping the raindrops off the head of the head of state and any of his invited guests.
But look at that Marine's face. I'll bet he's saying to himself something along the lines of "Talk as long as you like, you #%$@&*%!, I can stand here holding this thing, without changing hands, longer than you can play in the rain until someone with better sense comes along and takes you inside." Well, eitrher that or "Gee, I wonder if he'll start melting if I let a drop fall on his head? No, better not find out 'cause if he does some poor swabby will have to come clean the excrement off the walkway."
The rules were rexaled a few years ago to allow both enlisted and officers, not in formation, wearing Class A (dress uniform with uniform blouse worn)or Class B (dress uniform without uniform blouse - think "business casual" for values that mean no suit jacket/sports coat) uniforms - to carry umbrellas or small packages. They are supposed to carry one or the other in the left hand so that a salute can still be rendered. My mental picture is of an enlisted Marine, holding an umbrella, saluting a Marine officer, also holding an umbrella, with the fiollowing exchange: "Semper Fi, eh wot?" responded to with "Ooh rah! Raw-ther*."
* - you have to imagine a very upper-class posh accent