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You don't need the name of the parameter in order to use parametric_plot3d:

sage: sol = solve(eqns, [x,y,z])[0]
sage: parametric_line = [eqn.rhs() for eqn in sol]; parametric_line
[-r5, -r5 + 25, r5]
sage: parametric_plot3d(parametric_line, (-10, 10))

But still you can get it if you want (it's not "unusable in next steps" as you claimed):

sage: parameters = set(sum([list(eq.variables()) for eq in parametric_line], []))
sage: assert len(parameters) == 1
sage: parameter = list(parameters)[0]; parameter
r5