It was certainly different in each country.
In France as far as I can have an experience of school physicals which was rather common, the agreement of the parents was not really required before a physical examination and their presence was not asked since it was a group exam in elementary school as well as in middle schooll or in high school under 18 age.
We were a group of 15 boys to undress together in a locker room supervised by an adult person, mainly a male Pe teacher, and then examined one by one in the examination room by the doctor, mainly a woman, assisted by the school nurse and sometimes a student nurse.
There was always a weight-in and a check of the height ( in middle school, the weight and the height were even checked in the gym by the school nurse assisted by the Pe teacher at the beginning of each term, so three times by a school year) and a check of the spine and reflexes and an auscultation and a "turn and cough" test for hernia, All of that was a standard part of all school examinations and did not require at all an agreement of the parents.
They were mandatory examinations performed at school each year and it was useless to come with a form filled by a family doctor in his private'office after an examination. The parents knew that we would be examined that day by the school doctor because they received at home a letter to tell them that our presence at the school examination was required.
Of course, the eyes, ears and teeth and throat were also checked and blood pressure taken and the skin was visually inspected from head to toes, there was at middle and high school also a urine and blood tests. and often on an other day of the year an X ray inspection with a bus in the court.
Normally there was also an examination of the genitals including a palpation of the testicles, a check of all parts of the penis including the open hole and so on, and at high school a visual check of the butts and after bending over and spreading the butts a visual check of the anus but not a DRE ( the first time was for military service in a pre-draft center exammination)
At middle and high school, we were also asked to move our legs and arms and to walk to check the balance and gait and to bend over ... and to squat.
At high school, we were sometimes questioned by the school doctor if we were still virgin and seldomly about our masturbation practice. It was a little embarrassing but was a common use and no other boy could hear the answer
Nothing which could have seemed unfair and not appropriate.
The presence of a chaperone was never mentioned on the form since there was always a nurse and in case of need the Pe teacher could have been considered like a subrogate parent.
No one would have thought to complain about the lack of privacy since it was the custom at all schools, it was in a certain way inspired by the military methods for military examinations since we had to undress in a group either in briefs or stark naked and to wait our turn to be examined and the nudity was required either at the beginning or at least for the last part of the examination for check of spine and genitals exams and other intimate inspections but it seemed normal in the 60's and 70's.
No parents would have thought that it was inappropriate at least for boys.
Some of you said that it could be different for girls who had not to undress so much as the boys and had not so intimate examinations and that it was clear while reading some school forms for sport.
I was in boys only school, so I cannot say how it was in France for girls but I presume that it was rather the same as in your country.
The girl's students were supposed to be more shy towards nudity and their shyness was considered as fair on the reverse of that of boys, so they had certainly more privacy and were not asked to undress so much in group.
Today, it seemed as a double standard practice but it was normal in the past.
We must remember that a teenage boy was less frequently examined completely by a private doctor than a girl when she entered to puberty, so it was logical to examin more thoroughly the boy's students than the girls.