Some test examples for the parametrization algorithm:

f:=expand( (y-x)*(y-2*x)*(y-3*x)*(y-4*x) + x^3);
# Has only 1 singularity, so the parameter computation is trivial

f:=113/2592000*x-19/2592000*y+119/129600*y^2-2/405*y*x-271/129600*x^2+7/18*y^2*x
-17/18*y*x^2-10*y^3*x+35*y^2*x^2-50*y*x^3+24*x^4+307/180*x^3-1/20*y^3+y^4;

f:=25+1326*y*x+3740*x^3*y+3252*x^2*y+3582*y^3*x^2+4030*x^3*y^2+1476*x^2*y^4+102*
y^6*x+546*y^5*x+1590*y^4*x+244*x+184*y+518*y^2+720*y^3+854*x^2+1338*x^3+576*y^4
+1101*x^4+282*y^5+14*y^7+y^8+84*y^6+4770*y^2*x^2+2706*y^3*x+2646*y^2*x+18*x^7+
508*x^5+132*x^6+x^8+2264*x^4*y+2124*x^3*y^3+1716*x^4*y^2+738*y*x^5+354*y^2*x^5+
570*x^4*y^3+550*x^3*y^4+122*y*x^6+28*y^6*x^2+318*x^2*y^5+8*y^7*x+56*y^3*x^5+56*
y^5*x^3+70*y^4*x^4+28*y^2*x^6+8*y*x^7;
# Parameter computation does not take much time, but parametrization takes 500
# seconds on a fast machine. Probably evala(Resultant()) is not nearly as
# efficient as resultant().

f:=-89325/138014+413549/138014*y-185984/69007*y^2+1518117/552056*y^3-19259/69007
*y^4+y^5-155466/69007*x+153719/69007*x*y-392407/138014*x*y^2+31903/138014*x*y^3
-66141/69007*x*y^4-314712/69007*x^2+165666/69007*x^2*y-392063/276028*x^2*y^2-
50807/138014*x^2*y^3-81846/69007*x^3-50807/138014*x^3*y+244125/138014*x^3*y^2-
688135/552056*x^4*y;

f:=-y-4*y^2-y^2*x-5*y*x-2*x^2+3*x^2*y-6*y^3-2*y^4+x^3+3*x*y^4+6*y^5*x^2+4*x*y^7+12
*y^5*x+3*y^6+y^9+x^2*y^2+6*x^2*y^3+x*y^3-2*y^5+12*x*y^6+4*y^8+6*y^7+4*x^3*y^3+4
*y^2*x^3+12*y^4*x^2+x^4*y;