In Book 5, it states that being an Auror is the only profession Harry considered after leaving Hogwarts. I figured Harry was subconsciously following in his father's footsteps. It's odd to me that neither Lupin nor Sirius ever told Harry what his father's profession was. But Aunt Petunia was too quick to tell Marge that Harry's father was unemployed which suggests that James Potter did work for the Ministry of Magic. Also being an Auror would made James and his family a target for Voldemort.

Lily's profession as a healer still seems right to me. Healers are proficient in charms.

On the other hand, when Harry went into Olivander's for his wand, he first tried his father's wand and it blasted the drawers open. Next he tried his mother's wand and it broke a vase. Finally Harry tried Voldemort's wand and it was just right.
So you could argue that Harry will follow neither parents profession. But I believe that the wand chooses the wizard and the wizard chooses his profession.