I am not sure what the problem is. The following Manipulate updates both plots at the same time:
With[{x1 = t0, y1 = t0^2, l = 1, T = 1},
Manipulate[
Row[{Graphics[{{Thick, Line[{{0, l}, {x1, y1}}]}, {Red,
Disk[{x1, y1}, 0.2]}}, PlotRange -> {{-l, l}, {-l, 2 l}}],
Spacer[20],
ParametricPlot[{x1, y1} /. t0 -> a, {a, 0, T},
PlotRange -> {{-1, 1}, {0, 1}}]}], {t0, 0, 1, AnimationRate -> 1,
RefreshRate -> 60, Appearance -> "Open"}]]