That's common, because when the camera meter takes in the snow, it by design wants to turn the metered subject "medium", which makes the snow gray!
A good approach is to meter the snow, spot or center, whatever you've got to get snow in the center "properly" metered, then up the exposure by 1-2 EV stops. The safext way to do this and keep it on track is Manual, so you can set the exposure for the scene, then focus on the subject/bird and the camera "won't be fooled again".
Before "real" shooting, you might do a test shot and ensure from your display and histogram that the snow is up against the right of the histogram, but not blowing out. With this type of setting the birds will be more properly exposed.
The opposite problem can happen if the camera tries to brighten a dark bird and you end up blowing out the snow. You can take the same approach either way.
Note that if the scene lighting changes, you will want to compensate for that change, but you are smart to do this whether shooting Manual or a semi-auto setting such as Av.