Correct Age to appear 10th Std CBSE Board Exam

CBSE rule is for minimum age and maximum age. Private schools will have their own regulations and it is their decision to give admission to a candidate or not. There are schools that are making their own terms and conditions for admission within the guidelines given by the board. You have to pursue with the school administration in this regard.
As per the guidelines given by CBSE, the minimum age to appear in the 10th board exam is 14 years, as of 31st December of the year of examination. No upper limit of age to appear in the exam is specified. So the school will decide within these guidelines.
Your son is eligible to appear for the 10th examination in 2028. You have to contact the concerned officer in the school and pursue with him.

If you consider the norms of the private schools, they have the regulations of their own but still they follow the minimum age criteria stipulated by CBSE for the respective classes. As a thumb rule, the minimum age to appear class ten under CBSE pattern is 14 year as on 31 st December of the year of examination.
As stated by you, there are some schools framing their own rules as 15 years being minimum for writing class ten but the same yardstick does not hold good for all the schools. You might try the other one accepting 14 year as being the minimum age to write the examination.
Now talking to the case of your son, he would be eligible to write his class ten examination in 2028 when he attains the minimum age eligibility clause. You may chase the school authorities for the admission purpose keeping your view before them. I think they would oblige you.