I don't mean to sound unsympathetic but you're just going to have to get thicker skin. I work in a pediatric ER (not much different than urgent care or primary to be perfectly honest) and parents leave irate almost daily. The thing is, it's your job to do what's best for the child/patient even if they (parents!) don't agree. You don't have to be rude or unkind but you can't win them all! (and you won't!!)
Unnecessarily prescribing an antibiotic (or 'insert your patient's requested intervention here) is not only senseless, it's dangerous! A popular speech my attending uses frequently for viral pharyngitis R/T adverse effects of Amoxicillin: 1/10 kids will develop a rash, 1/25 kids will develop diarrhea, and 1/1000 kids will develop a severe, life-threatening reaction to the abx. Now compare that to the risk of rheumatic fever: 1/1,000,000!!! There is less chance of developing this life-threatening illness than of the patient developing a reaction to Amoxicillin!! (you'd have to double check my stats but I believe they are pretty accurate)
Now enough of my rant--the moral of the story is to stick to your guns. The best thing you can do is keep up with your reading so you can present the most up-to-date evidenced-based reasoning for your plan of care. But please, please, please don't practice defensive medicine--it can be just plain dangerous.
That said--hope things get better. I'm sure you're doing great