I'm sure it is, depending on just how long this re-route takes. I should hope that it's obvious that if they can simply steer around an illegally parked car without any issue, they should just do this, for many reasons. But if the double/triple-parked cars are completely blocking the road, they can plow through, and they have every right to.
A single double-parked car in my city can result in the street being completely inaccessible for thru-traffic. I've also seen them break out the spotters and back up into their turning lane.
I assume they're also reticent to implement the nuclear option, because even a cursory google search didn't return any incidents of this happening anytime recently (though obviously that doesn't mean it doesn't).
It did, however, return a few results about fire trucks being in (non head-on) collisions on city streets and several firefighters being injured on-board.
When a fire truck pushes another vehicle out of the way, it doesn't ram into it at full speed; that would be dumb. It slowly makes contact with the vehicle, then pushes. A fire truck easily has enough power to push a parked car out of the way. Of course, this doesn't happen often.