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I am afraid you can't do a 3D parametric plot with Piecewise.

I suggest to use a python dict to define the functions, for example:

F = {(-2,-1):[t,e^-1,0], (-1,1):[t,e^-t^2,0], (1,2):[t,e^-1,0]}
G = Graphics()
for k in F.keys():
    G += parametric_plot3d(F[k],k)
G.show()