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Sage does not try to automatically put elements in the smallest subfield that they belong to. You need to do that explicitly. You can use something like the line g = P(f) below which converts an element f from GF(103^2) to an element g of GF(103) :
sage: R.<a> = GF(103^2)
sage: P.<b> = GF(103)
sage: sage: f = R(99); f
99
sage: f.parent()
Finite Field in a of size 103^2
sage: g = P(f); g
99
sage: g.parent()
Finite Field of size 103
