The developers recommend that professionals enlist parents to complete the questionnaires whenever possible. ASQ was developed and validated as a parent-completed tool, and many studies have found parents to be reliable evaluators of children’s behavior. As you’ve experienced, there are some scenarios—such as a cognitive disability, limited reading skills, or active drug abuse—that can make it difficult for parents to complete the questionnaire on their own. In these cases, parents will need support in the process. Some parents may only need minimal assistance, such as clarification on a particular item, while others may need a great deal of support (for example a parent with a cognitive disability may need support reading items, trying them out with their child and interpreting their child’s responses).
If a professional will be completing the questionnaires, guidelines require that they have at least 15-20 hours of contact with the child each week, for at least a month. This ensures that the person completing the questionnaire is familiar with the child's skills. It doesn’t sound like your staff members have that level of familiarity with the child, so it is not recommended that staff members complete the questionnaire independently.
It would be advisable to have the staff member complete the questionnaire (or portion of the questionnaire that caused an issue) with the parent during the appointment. The ASQ authors offer guidance for using questionnaires during home visits that can be helpful in your situation. Details for ASQ-3 are on pages 112–114 of the ASQ-3 User’s Guide. Instructions for ASQ:SE-2 are on pages 140–142 of the ASQ:SE-2 User’s Guide.