From this site:

  eqs = {x == 1 - t^2, y == -2 t - 2};
  eq2 = {x, y} /. (And @@ eqs // ToRules)
  ParametricPlot[eq2, {t, -5, 5}, MeshStyle -> Red, Mesh -> True, Epilog -> Arrow@Partition[Table[eq2, {t, -5, 5, 0.5}], 2]]

Or, from:

   ParametricPlot[eq2, {t, -5, 5}] /. Line[x_] :> Sequence[Arrowheads[Table[.05, {10}]], Arrow@Line[x]] 

It should be an option to ParametricPlot[]. But here's another easy workaround.

f[t_] := {Cos[t], Sin[t]}
Show[
 ParametricPlot[f[t], {t, 0, 3/5 Pi}], 
 Graphics[{Red, Arrowheads[.1], 
   Arrow[{f[3/5 Pi - .01], f[3/5 Pi]}, -.02]}]]

This is done with labels:

f[t_] := {Cos[t], Sin[t]}
ParametricPlot[f[t], {t, 0, 3/5 Pi},
 Epilog -> Table[{PointSize[Medium], Point[f[s]],
    Text[t == s, f[s],
     1.3 RotationMatrix[Pi/2] . Normalize[f'[s]]]},
   {s, Subdivide[0, 3/5 Pi, 5]}],
 PlotRangePadding -> Scaled[.15]]