Two functions both change with the same independent variable t, there is another variable represent different groups (z). how to plot y[t] versus x[t] with each group of z represented by one independent curve. See example equation below, with z as a separate integer variable that varies while t is a continuous variable. Please note I do not plan to use Manipulate function.
x[z_][t_] := z/5*Exp[-0.1*t]
y[z_][t_] := 500/(2 + x[t])
z={1,2,5,10,20}
ParametricPlot[{Evaluate[x[z][t]],Evaluate[y[z][t]},{x,0,100},{z,zlist},AspectRatio -> 1]