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When evaluating a univariate polynomial, it correctly takes the parent of the input into account:
sage: k.<a> = GF(2^4)
sage: R.<x> = PolynomialRing(k)
sage: f = R(1)
sage: S.<y> = PolynomialRing(k)
sage: f(y).parent()
Univariate Polynomial Ring in y over Finite Field in a of size 2^4
Multivariate polynomials do not:
sage: k.<a> = GF(2^4)
sage: R.<x> = PolynomialRing(k, 1)
sage: f = R(1)
sage: S.<y> = PolynomialRing(k, 1)
sage: f(y).parent()
Finite Field in a of size 2^4