i'm not sure if ...彼女の絵 is saying "her drawing that she got as a birthday present (from her father)" or "the drawing she drew that her father got as a birthday present".
Given how the first part of this is extolling her brilliance as a musician it would only be fitting to also suggest her mastery of other arts, but the しかも start of the 3rd sentence makes it hard to absolutely assume that.
Since に marks the source of something is received from but is marking 前の誕生日プレゼント in this case, i find it awkward to rearrange this relative clause.
the ambiguity of 父が有名な画家で絵をたしなむ make it harder to decifer.
if someone is a "有名な画家", can they be someone only "has a taste for painting"?