In[]:=
f[t_] := t + 1
g[t_] := t^3 + 3*t + 12
ParametricPlot[{f[t]*Cos[θ], f[t]*Sin[θ]}, {t, 0, 1}, {θ, 0, 2*Pi},
ColorFunction -> Function[{x, y, t}, ColorData["SolarColors"][Rescale[g[t], {g[0], g[1]}, {0, 1}]]]]
g[t_] := t^3 + 3*t + 12
ParametricPlot[{f[t]*Cos[θ], f[t]*Sin[θ]}, {t, 0, 1}, {θ, 0, 2*Pi},
ColorFunction -> Function[{x, y, t}, ColorData["SolarColors"][Rescale[g[t], {g[0], g[1]}, {0, 1}]]]]
Out[]=
