I cannot recall a built-in method to reverse an axis, at least for ParametricPlot, but maybe the right FrameTicks syntax will help:

ticks = {{{-6, 6}, {-5, 5}, {-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}, {0, 0}}, {{0, 
   0}, {1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}}, {}, {}};

ParametricPlot[
 spiral[.002*#^(5/3), t, -#, #] & /@ Range[objects], {t, 0, 10*\[Pi]},
  PlotRange -> {{-objects - 1, 0}, {0, objects + 1}}, 
 PlotLabel -> Style["3. x axis reversed\nticks lost", FontSize -> fs],
  Frame -> True, FrameLabel -> fl, GridLines -> Automatic, 
 FrameTicks -> ticks]

It's possible that this version-8 function may work with ParametricPlot, though I can't test that, and ParametricPlot doesn't appear to be supported: ScalingFunctions

Update

In Mathematica 10 ScalingFunctions does work with ParametricPlot, though it is undocumented.

Table[
  ParametricPlot[spiral[.002*#^(5/3), t, #, #] & /@ Range[objects], {t, 0, 10*Pi}, 
  GridLines -> Automatic, ScalingFunctions -> sfn, PlotLabel -> {sfn}],
 {sfn, 
    {{Identity, Identity},
     {Identity, "Reverse"},
     {"Reverse", Identity},
     {"Reverse", "Reverse"}}
 }
]

I cannot recall a built-in method to reverse an axis, at least for ParametricPlot, but maybe the right FrameTicks syntax will help:

ticks = {{{-6, 6}, {-5, 5}, {-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}, {0, 0}}, {{0, 
   0}, {1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}}, {}, {}};

ParametricPlot[
 spiral[.002*#^(5/3), t, -#, #] & /@ Range[objects], {t, 0, 10*\[Pi]},
  PlotRange -> {{-objects - 1, 0}, {0, objects + 1}}, 
 PlotLabel -> Style["3. x axis reversed\nticks lost", FontSize -> fs],
  Frame -> True, FrameLabel -> fl, GridLines -> Automatic, 
 FrameTicks -> ticks]

It's possible that this version-8 function may work with ParametricPlot, though I can't test that, and ParametricPlot doesn't appear to be supported: ScalingFunctions

Update

In Mathematica 10 ScalingFunctions does work with ParametricPlot, though it is undocumented.

spiral[a_, t_, x_, y_] := {a*t*Cos[t] + x, a*t*Sin[t] + y};

ParametricPlot[
 spiral[.002*#^(5/3), t, #, #] & /@ Range[5]
 , {t, 0, 10*Pi}
 , GridLines -> Automatic
 , ScalingFunctions -> #
 , PlotLabel -> {#}
] & /@ Tuples[{Identity, "Reverse"}, 2]

I cannot recall a built-in method to reverse an axis, at least for ParametricPlot, but maybe the right FrameTicks syntax will help:

ticks = {{{-6, 6}, {-5, 5}, {-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}, {0, 0}}, {{0, 
   0}, {1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}}, {}, {}};

ParametricPlot[
 spiral[.002*#^(5/3), t, -#, #] & /@ Range[objects], {t, 0, 10*\[Pi]},
  PlotRange -> {{-objects - 1, 0}, {0, objects + 1}}, 
 PlotLabel -> Style["3. x axis reversed\nticks lost", FontSize -> fs],
  Frame -> True, FrameLabel -> fl, GridLines -> Automatic, 
 FrameTicks -> ticks]

It's possible that this version-8 function may work with ParametricPlot, though I can't test that, and ParametricPlot doesn't appear to be supported: ScalingFunctions

I cannot recall a built-in method to reverse an axis, at least for ParametricPlot, but maybe the right FrameTicks syntax will help:

ticks = {{{-6, 6}, {-5, 5}, {-4, 4}, {-3, 3}, {-2, 2}, {-1, 1}, {0, 0}}, {{0, 
   0}, {1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}}, {}, {}};

ParametricPlot[
 spiral[.002*#^(5/3), t, -#, #] & /@ Range[objects], {t, 0, 10*\[Pi]},
  PlotRange -> {{-objects - 1, 0}, {0, objects + 1}}, 
 PlotLabel -> Style["3. x axis reversed\nticks lost", FontSize -> fs],
  Frame -> True, FrameLabel -> fl, GridLines -> Automatic, 
 FrameTicks -> ticks]

It's possible that this version-8 function may work with ParametricPlot, though I can't test that, and ParametricPlot doesn't appear to be supported: ScalingFunctions

Update

In Mathematica 10 ScalingFunctions does work with ParametricPlot, though it is undocumented.

Table[
  ParametricPlot[spiral[.002*#^(5/3), t, #, #] & /@ Range[objects], {t, 0, 10*Pi}, 
  GridLines -> Automatic, ScalingFunctions -> sfn, PlotLabel -> {sfn}],
 {sfn, 
    {{Identity, Identity},
     {Identity, "Reverse"},
     {"Reverse", Identity},
     {"Reverse", "Reverse"}}
 }
]