"It wasn't the Speaker's car. Comment is free but facts are sacred."
Hmmm! I wonder who the car is registered to?

In my view the point is irrelevant, In his role as speaker Bercowis expected to show decorum and impartiality in all aspects of his public life.
Many would see a car belonging to the Bercow family with a bollocks to brexit sticker attached to be not simply raising two fingers to those that voted leave, but a direct display of his partisanship. If he and his wife are not capable of making that connection, then perhaps the speaker is unsuited to his role.The sticker incident is but one of numerous actions where his impartiality can be questioned.
I am afraid to excuse it as being purely the actions of a totally independent person, namely his wife, does not diminish the affront, or his demeaning of his office.