We use FrenetSerretSystem to calculate its tangent vector and normal vector at c[t0].
c[t_] = {t^2, t};
tangent[t_] = FrenetSerretSystem[c[t], t][[2, 1]];
normal[t_] = FrenetSerretSystem[c[t], t][[2, 2]];
s[t_] = ArcLength[c[τ], {τ, 0, t}];
t0 = .5;
Show[ParametricPlot[c[t], {t, 0, 1}],
Graphics[{Arrow[{c[t], c[t] + normal[t]}],
Arrow[{c[t], c[t] + tangent[t]}]}] /. t -> t0, PlotRange -> All]
And then we calculate the ArcLength of the parametric curve {t^2,t} from 0 to t0 then we get the point {0,s[t0]} in y-axis.
After that we translate {0, s[t0]} - c[t0] and rotate tangent[t0] to the y-axis direction {0,1} so that the new curve tangent to the y-axis at {0,s[t0]}.
c[t_] = {t^2, t};
tangent[t_] = FrenetSerretSystem[c[t], t][[2, 1]];
normal[t_] = FrenetSerretSystem[c[t], t][[2, 2]];
s[t_] = ArcLength[c[τ], {τ, 0, t}];
r[t_, t0_] :=
RotationTransform[{tangent[t0], {0, 1}}, {0,
s[t0]}][(c[t] + {0, s[t0]} - c[t0])];
Manipulate[
ParametricPlot[r[t, t0], {t, 0, 1}, AspectRatio -> Automatic,
PlotRange -> {{0, 1}, {0, 2}}], {t0, 0, 1}]
