Child travelling without parents

Not needed as far as I know. I went to boarding school far away from my home country (think like in a totally different continent separated by an ocean and I needed to travel to an intermediate continent to get to my home continent) and began travelling alone on planes at the age of 13. No special form needed in my case and I travelled like that year after year throughout the rest of my childhood.

That said, checkout the "unaccompanied minor" page of the specific airline your child will be travelling on. For example, this is United's: https://www.united.com/en/us/fly/travel/accessibility-and-assistance/unaccompanied-minors.html

When I was a child, I typically flew on KLM to return home. KLM's page seems similar to United from just skimming it: https://www.klm.com/information/assistance-health/children-travelling-alone

For Spain yes you do, I'm not sure about other countries. Not only would I provide a notarized travel consent form I would also do a medical consent form and copies of the minor's insurance forms signed by both parents and a copy of the birth certificate. When my grandson was a minor and we traveled around the USA (and to Europe) I always had those documents with me. Forty plus years ago my mother took my daughter to the emergency room because she was having difficulty breathing, they would not treat my daughter until I arrived. So the moral of the story is: play it safe. J

And pay attention to what is considered "minor"; that is, to the AGE. For example, look at Turkish airlines: https://www.turkishairlines.com/en-int/any-questions/infants-and-children/unaccompanied-child-passenger/

Accompanied minor passenger: Passengers aged 2 and under 7 years old
may be accompanied by a cabin crew member (for a fee) during the
flight at the request of their parent or the person authorized by
their parent.

Unaccompanied minor passenger: Passengers aged 7 and under 12 years
old may travel alone if a written authorization is provided by their
parent or legal representative (Unaccompanied Minor Passenger Carriage
Authorization). The parent or legal representative must wait at the
airport until the flight has departed.

For your child who is 15, none of the above would seem to apply.