The function definition of f and g do not match the call.
Something like this seems to work
f[a_, r_, t_, v_, w_] := v t (1 + (1/(1 - (r/a) Cos[w t])))
g[a_, r_, t_, v_, w_] := (r Sin[w t])/(1 - (r/a) Cos[w t])
Manipulate[
ParametricPlot[{f[a, r, t, v, w], g[a, r, t, v, w]}, {t, 0, 10}],
{r, 0, 1},
{a, 1, 20},
{v, 0, 20},
{w, 0, 20}
]
