If a parent is not able to answer certain questions at the time of the screening, is it permissible for the teacher to observe the child in a classroom setting and answer the missed questions at a later date?

It's not ideal to have different people complete different items on the same exact questionnaire. But if skills aren't seen in one specific setting, then input from other adults, such as teachers, might be helpful. Some programs have the parent and teacher complete separate questionnaires and discuss their answers. It is always ideal if parents and teachers are working together, and it may be that input from both a parent and teacher is needed to answer ASQ-3 items for some children.