Unless I'm misreading, I think that's what
@ThatNaiGuy is referring to -- that they could use Parla's lower-stakes procedure to check how healing magic interacts with a fresh operation. And if all goes well, then they could even consider using it on the King's hand.
It'd be a good idea, but they would need consent from the patient for it first - and since it's a pure cosmetic surgery on the face the odds of her being willing to risk it isn't super high. If they have to redo it all because it caused another bad scar the next surgery will be larger and the result more noticeable.
Asking someone to let them run this experiment on something like a leg or other place that won't be seen much is probably a better idea.
What I want to know on the other hand, is that since healing magic seems pretty tied to the magician's understanding of the internals of what they're healing, if this could be a learning experience for her to be able to heal with less scarring. A mental image of suturing wounds up before healing the original wound could maybe already have made it a lot smaller the first time.
So many interesting experiments to run...