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There is undocumented "CoordinatesToolOptions" -> Identity which specifies the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

If displaying x, y, t is required then definition of fu is:

fu = Function[{x}, {Splice@x, 
    Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

New version: (after @Domen's useful comment)

fu1 = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 
     2 Pi]}];

fu2 = Function[{x}, {Splice@x, 
    Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

(* tooltip t *)
ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}, 
 CoordinatesToolOptions -> {"DisplayFunction" -> fu1}, 
 ImageSize -> Small]

(* tooltip x, y, t *)
ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}, 
 CoordinatesToolOptions -> {"DisplayFunction" -> fu2}, 
 ImageSize -> Small]

Old version:

There is undocumented "CoordinatesToolOptions" -> Identity which specifies the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

If displaying x, y, t is required then definition of fu is:

fu = Function[{x}, {Splice@x, 
    Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

There is undocumented "CoordinatesToolOptions" -> Identity which specify the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

If displaying {x,y}, t is required then definition of fu is:

fu = Function[{x}, {x, 
   Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}]

There is undocumented "CoordinatesToolOptions" -> Identity which specifies the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

If displaying x, y, t is required then definition of fu is:

fu = Function[{x}, {Splice@x, 
    Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

There is undocumented "CoordinatesToolOptions" -> Identity which specify the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

There is undocumented "CoordinatesToolOptions" -> Identity which specify the function applied to coordinates {x, y}. Unfortunately I could not figure out how to set it straight within ParametricPlot so I had to use awkward method with ToString, FullForm, StringReplace and ToExpression.

Another issue is that it seems to be quite hard to figure out the definition of function fu that transforms {x, y} into parameter t.

But if all is done correctly it works nicely like the default behaviour.

pl = ParametricPlot[{Cos[t], Sin[2 t]}, {t, 0, 2 Pi}];

spl = ToString@FullForm@pl;

fu = Function[{x}, {Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}];

ToExpression[
 StringReplace[spl, 
  "Rule[\"CoordinatesToolOptions\", Identity]" -> 
   ToString@FullForm@Rule["CoordinatesToolOptions", fu]]]

If displaying {x,y}, t is required then definition of fu is:

fu = Function[{x}, {x, 
   Mod[2 ArcTan[x[[1]] (1 + x[[1]]), x[[2]]/2], 2 Pi]}]