Then we need to start studying how to help parents have a better experience when vaccinating their children. The fact is that, as a parent, it's unsettling to watch a doctor pump vial after vial into your kid. Even immediately after birth, I remember there was a big vial they injected into my son. If the patient had been an adult it would have been like injecting an entire tumbler full of serum. There's a bit of a gut punch there, especially if you're like me and generally believe that you were born with pretty much everything you needed to live well and that seeking medical treatment is something you do because you can't otherwise live well.
I'm good with vaccines, but we shouldn't be dismissing the parental perspective. Also, studied side effects probably don't include things like increased agitation, crankiness, lethargy, as side-effects. These things make life miserable for parents for a few days so yeah, parents are going to notice. Of course some parents are going to blame the vaccine when the kid just happened to get a cold at the same time too, but people get the message of "trust your instincts" and then see their kid get sick, and it is the parent that is ultimately responsible for the child's well being and also the one that will be blamed if the child is having a bad day or year.
So yeah, some amount of working with parents would be a good thing.