It would be legal to drive if it otherwise conformed to the rules about street-legal cars. Each state has a list of requirements that have to be met in order for a vehicle to be legal to drive. Since the video takes place in Florida, I took a glance at Florida's regulations.

The biggest two are:

Lights: The front lights are fine, but whichever end of the vehicle is designated as the "back" would need two red taillights, a white license plate light, and two red stoplights.

License plate: There has to be a license plate on the back, and possibly the front. If it were issued two plates, it would be the same plate (it's one vehicle).

What they don't really show in the video is modifying the actual driving parts of the car - they show cutting the two cars apart and welding them together, but they also show the car "driving" into the spot. You can't just cut a transmission in half, glue two "fronts" together, and get a car that drives. There's work that has to be done to the drive train, brakes, and transmission (at least) just to get the car to move "forward". It could be possible to make a car that is driveable from both ends, but that would require more engineering than our Youtubers put in.

So, what you'd really wind up with is a normal car with the backseat, trunk, and rear body removed, but not touch the chassis. You then stick the body of another car "backwards", leaving out all the actual under-the-hood bits.

So you wind up with a car with an actual "Front" and "back", if for no other reason than, as pointed out with the lights, there are actual laws about what sorts of lights you must have on the car and where those lights need to be.

And no, I doubt that would actually get you out of the "Back end parking ticket" as the car does, actually, have a 'back end'.