ISPOR provides "good research practices for the assessment of children and adolescents." See Matza, L. S., Patrick, D. L., Riley, A. W., Alexander, J. J., Rajmil, L., Pleil, A. M., & Bullinger, M. (2013). Pediatric patient-reported outcome instruments for research to support medical product labeling: report of the ISPOR PRO good research practices for the assessment of children and adolescents task force. Value in health : the journal of the International Society for Pharmacoeconomics and Outcomes Research, 16(4), 461–479. doi.org/10.1016/j.jval.2013.04.004

Because pediatric self-report and parent proxy report are not equivalent, data from these measures should be analyzed separately. That is, if you have a sample where half of the participants completed a self-report measure of anxiety and half had a parent-proxy report of anxiety, don't lump all anxiety scores together and ignore who provided the data.

Pediatric self-report is considered the standard for measuring patient-reported outcomes among children. For proxy report, circumstances do exist when the child is too young (as you note, below age , cognitively impaired, or too ill to complete a patient-reported outcome instrument. Self- and proxy- report are not identical and you should be cautious about treating scores that way.

You could collect both self- and proxy- report for some of your sample if you wanted to evaluate the difference between the measures. While information derived from self-report and proxy-report is not equivalent, it is optimal to assess both the child and the parent since their perspectives may be independently related to healthcare utilization, risk factors, and quality of care.

Darren Dewalt has done considerable work in the use of PROMIS in pediatric populations. You can look into his work further to see if he's published papers comparing scores between self- and proxy report.