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]
```

[![enter image description here][1]][1]

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}]
```

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/TtQk6.png
  [2]: https://i.sstatic.net/d4VUl.gif