I stand to be corrected, but yes, I think she does. I think she has a responsibility to promote (in the very least) and encourage the safety and well being of pupils on their way to school. She also has a moral responsibility (again at the very least) to manage the school as part of the wider community.
She probably has quite an extensive remit as far as "parking issues" are concerned. I know for instance that double parking and or parking on the chevrons outside school can restrict access for emergency services, it blocks the view of people coming out of the drop off zone.
So whilst it may not have been "illegal" it must have been a cause for concern and yes, I think she absolutely has a right to send a photograph around the (limited) public which constitutes your school.
And to be quite frank - taking issue with the way the head has dealt with it, rather than perhaps going in to see her to explain it was a one off I remain to be convinced and apologising profusely, then going to see the neighbour with a bouquet of flowers, seems rather odd.
And when you say "I couldn't see what else I could do" - there are a million and one suggestions - get out of bed earlier / park further away and walk in / you wait until a space becomes free and then park even if that means your child will be late / you share lifts so there aren't as many cars. You never, ever block a neighbours drive. What if they had a hospital appointment? Needed to get their own children to school??