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If you want to compute the projection of p2 in S, you should write:
sage: q = S(p2)
sage: q
2*ybar^2
sage: q
2*ybar^2
sage: q.parent()
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x - y)
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x - y)
In a general quotient ring, the degree is not well defined, but you can get the minimal degree of a representative of q in R by lifting q:
sage: q.lift().degree()
2
