As noted by someone above, we screened kids on at minimum 9 instruments and up to all 11. There must be a parent present. The parent can witness firsthand their student struggling to make a sound on the alto saxophone or having any kind of struggle with coordination or rhythmic aptitude on a drumpad is usually all it takes. They can also see what they did really good at. Further, the director can be there to celebrate and push them positively towards something and make them feel really good about it!
Make it all about that you have their student's success as your #1 priority. That lets parents know you're in their corner.
Beyond that, we explain it out in sports terms / analogy. (People seem to understand those) Band is like a football team *cringe* you gotta have all the positions covered or the team won't be successful. The defensive lineman is just as important as the quarterback. We also phrase it (politely, of course) something like this to the kid/family - Do you get to walk up to the football coach and tell them what position you're going to play?
Finally, sometimes it also comes down to what you call your instrument selection night. Do you call it a "try out" an "instrument fitting" or a "selection night". Sometimes that sets the tone.