Brown or red, the color doesn't really matter - It's the geup (grade level). Some Taekwondo schools use brown for the high belt, some use red. Some use both, with brown before red, and others have red before brown. Also, many schools have a temporary Black Belt stage between color belt and Black Belt (particularly for adults). This level often uses a half brown/half black belt, or a plain black belt with no degree stripe. It is sometimes called "Recommended Black," "Deputy Black," or "Black Belt Candidate." This can last six months to one year. The Poom belt for children often takes the place of this until the child reaches age of majority and can be awarded a 1st Dan.
In any case, on average, the student spends two to three months in the lower grades, four to six months in the higher grades, and usually six months to one year in the final grade before the 1st Dan. A student with prior experience, or an exceptional student might have any of the time periods reduced by a month or two, or skip a grade (color of belt) or two.
The Chief Instructor of the Dojang is the only one who should make this kind of recommendation, and if they are part of a larger organization (which issues the certificate rather than a school certificate), the Kwan Jang would probably need to approve an early promotion to Black Belt.
Overall, the entire time from white belt to First Dan should never, under any circumstances, be less than two and half years (which would include any previous training time at another school). Three to five years is ideal, and it could very well take longer. Keep in mind, a thirteen year old is going for Poom and not a Dan rank (even if your school does not recognize poom, no child at the age of thirteen is fully developed to hold the same level as an adult Black Belt (I'm not particularly fond of the 15 year old limit for Kukkiwon). However, this promotion will give your son time to develop as a poom before maturing in the Art, and in life, to become a full Black Belt Dan at the appropriate age.
My recommendation would be to discuss this with your son, and the Chief Instructor, then come to a sound decision as the parent. I see no problem with the time requirement as you have described.
CM D. J. Eisenhart