That is an interesting approach. It is similar to the parallelogram already in widespread use in bicycle rear derailleurs. But, they differ in that the derailleur receives force applied directly to one of the three moving sides of the parallelogram (one side is fixed to the bike frame). So, the derailleur force is applied very close to the critical pivot connecting one of the moving sides to the fixed side. On the other hand, when used as a piano key, the force would be applied to the key at a great distance from that critical pivot point. What effect these differences would have is another discussion.
I would anticipate that a main weakness would be the necessity for four pivots, one at each corner of the parallelogram. These corner pivots are the main point of destructive wear in a bicycle derailleur. As they wear, the parallelogram can begin to operate out-of-plane as the forces applied to the initial side can move the next, subsequent side in a twisting or direction different from its originally intendeded direction.
Though pivot wear is similarly a factor in plastic piano actions, it is not a factor in an acoustic piano that uses a balance rail instead of a pivot.
Another weakness of the parallelogram would be the necessity for a spring or other unnatural force to return the key after is it released.
If available space was not an issue, this could perhaps be mitigated by connecting the parallelogram to the piano at the midpoints of the horizontal sides of the parallelogram, and then adding appropriate counterweight to the side of the parallelogram that is exactly opposite the side to which the key is attached. It seems like this could introduce a more 'natural' feeling resistance force to the key stroke.
I'm out of time and have to run.
This parallelogram thing is interesting.