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In case of an existing rational parametrization of the curve you need only one parameter, call it t - not two parameters, u, v as in the question. (A curve is something with dimension one.) So in our case:

R.<x,y> = PolynomialRing(QQ)
C = Curve(2*x^5 + x^2*y - 4*x^3*y + 2*x*y^2 + 2*x*y^3 + 1*y^5)    # please use spaces around the = sign
par = C.rational_parameterization()    # each time i type parametrization instead...

Now we can ask for par.domain() and par.codomain()

sage: par.domain()
Affine Space of dimension 1 over Rational Field
sage: par.codomain()
Affine Plane Curve over Rational Field defined by 2*x^5 + y^5 - 4*x^3*y + 2*x*y^3 + x^2*y + 2*x*y^2

There are many other things in there. But you want the parametrization, computed in a point $t$ (instead of a tuple / double (u, v)). We take this t as it comes and ask...

sage: par[0]
(-2*t^2 - t)/(4*t^5 + 1)
sage: par[1]
(4*t^4 + 2*t^3)/(4*t^5 + 1)
sage: list(par)
[(-2*t^2 - t)/(4*t^5 + 1), (4*t^4 + 2*t^3)/(4*t^5 + 1)]
sage: par(17)
(-85/811347, 49130/811347)

So the list conversion of the object par gives already the components. Note that t is not really defined, the print just involves it. So if we want to have the components in "our" (other) special ring with unknown / transcendental variable $T$...

sage: S.<T> = PolynomialRing(QQ)    # or use some t from somewhere else
sage: u, v = par
sage: u(T)
(-1/2*T^2 - 1/4*T)/(T^5 + 1/4)
sage: v(T)
(T^4 + 1/2*T^3)/(T^5 + 1/4)