Use ContourPlot data in ParametricPlot

Update: Adding multiple mesh lines and legends:

mesh = {{9, Directive[Red, Opacity[1], Thick]}, 
        {10, Directive[Green, Opacity[1], Thick]}, 
        {12, Directive[Black, Opacity[1], Thick]}};

ParametricPlot[{u t, u^2 + t^2}, {u, 0, 10}, {t, 0, 10}, 
  MeshFunctions -> {Function[{x, y, u, t}, u + t]}, 
  Mesh -> {mesh}, 
  PlotLegends -> LineLegend[## & @@ Reverse[Transpose[mesh]], LegendLabel -> "u + t"]]

Original answer:

You can use the argument of ContourPlot as the MeshFunctions option value in ParametricPlot as follows:

ParametricPlot[{u t, u^2 + t^2}, {u, 0, 10}, {t, 0, 10}, 
 MeshFunctions -> {Function[{x, y, u, t}, u + t - 10]}, 
 Mesh -> {{0}}, 
 MeshStyle -> Directive[Red, Thick]]