You are using the same "t" inside ParametricPlot and as the animation variable. Use "t" inside ParametricPlot and e.g. "s" to animate. (x, y and z are of no use). Try this code:
Animate[ParametricPlot3D[{Sin[t + s], Cos[t + s], t}, {t, 0, 5 Pi},
PlotRange -> {{-1, 1}, {-1, 1}, {0, 5 \[Pi]}}], {s, 0, 2 Pi}]
