Unfortunately, version dependent solution is not good enough for me.

Okay.

I was hoping for a more elegant solution.

And your hope comes true ... you program it along the following lines

In[34]:= Clear[arrowParametricPlot]
arrowParametricPlot[f_List, p_List] := Block[{[Delta] = (p[[3]] - p[[2]])/100.},
   ParametricPlot[f, p, Epilog -> {Arrowheads[.1, 0], Arrow[{f /. p[[1]] -> p[[3]] - [Delta], f /. p[[1]] -> p[[3]]}]}, PlotRange -> All]
   ] /; Length[f] == 2 &&  Length[p] == 3 && !FreeQ [f[[1]], p[[1]]] && !FreeQ[f[[2]], p[[1]]]

producing pictures like

of course, there's no free lunch, so get the color of the arrow right on your own, please.

Thank you, this worked to some extent, but not in all the cases I need it. For example in

Graphics[Arrow[ParametricPlot[{Cos[t], Sin[t]}, {t, 0, -1}][[1, 1, 3, 2]]], Axes -> True]

the arrow goes to the wrong end of the curve. It goes to the end corresponding to t=0, not to t=-1. Is there some place I can look up a documentation how the FullForm of ParametricPlot is generated?

Unfortunately, version dependent solution is not good enough for me. I am trying to come up with some demonstrations to illustrate high school mathematics concepts (in this case the argument of a complex number), and it would not be acceptable to write a demonstration that works differently (or not work at all) a year from now. I will then simply put an arrowhead to the appropriate place by calculating the position. I was hoping for a more elegant solution. To be honest I don't really see a reason behind this implementation of ParametricPlot. Why is the FullForm output the same if the limits of the parameter interval change? Surely the starting and end values of the parameter interval have some meaning, even though the output is the same if we change them. In any case thank you for your help.