Create dynamically Manipulate buttons to come back to current parameters

Question

I would like to create dynamically buttons in Manipulate in such a way that after clicking the button the Manipulate parameters would come back to the values when the button was created. In my example after pressing "Save params" button I would like to add new button to Row with thumbnail of ParametricPlot with current set of parameters.

Manipulate[
 ParametricPlot[{Sin[a*u], Sin[b* u]}, {u, 0, c Pi}],
  Row[{
   Button[ParametricPlot[{Sin[2*u], Sin[3* u]}, {u, 0, 2 Pi}, ImageSize -> {30, 30}, Axes -> False], a = 2; b = 3; c = 2, ImageSize -> {50, 50}],
   Button[ParametricPlot[{Sin[5*u], Sin[6 u]}, {u, 0, 2 Pi}, ImageSize -> {30, 30}, Axes -> False], a = 5; b = 6; c = 2, ImageSize -> {50, 50}]
  }],
 {a, 1, 10},
 {b, 1, 10},
 {c, 1, 10},
 Button["Save params"]
]

Answer 1

Here's my go at it. I used a SetterBar with a Dynamic list of presets, since it implements a dynamic row of buttons for me. By not assigning a value to set, the button won't be highlighted, but clicking the button will have the desired action.

Manipulate[
 ParametricPlot[{Sin[a*u], Sin[b*u]}, {u, 0, c Pi}, PlotRange -> 1],
 {{presets, {{2, 3, 2} -> 
     ParametricPlot[{Sin[2*u], Sin[3*u]}, {u, 0, 2 Pi}, 
      ImageSize -> {30, 30}, Axes -> False],
    {5, 6, 2} -> 
     ParametricPlot[{Sin[5*u], Sin[6 u]}, {u, 0, 2 Pi}, 
      ImageSize -> {30, 30}, Axes -> False]}},
  ControlType -> None},
 {set, Dynamic@presets, SetterBar, 
  TrackingFunction -> (((*set=*){a, b, c} = #) &)},
 {a, 1, 10}, {b, 1, 10}, {c, 1, 10},
 Button["Save params", 
  AppendTo[presets, {a, b, c} -> 
    ParametricPlot[{Sin[a*u], Sin[b*u]}, {u, 0, c Pi}, PlotRange -> 1,
      ImageSize -> {30, 30}, Axes -> False]]]
 ]