Given the clarification from the edit, here are a few less severe options:
- worrisome — tending to worry: having a tendency to worry
- tender — with gentle feeling: showing care, gentleness, and feeling
- with or using "kid gloves"
- precautious — using or displaying precaution
Here are two relevant terms that have some relation to your intended meaning:
helicopter parent — a parent who pays extremely close attention to a child's or children's experiences and problems, particularly at educational institutions
Helicopter parents are traditionally seen as overly worrisome or paranoid. This could certainly apply to medical concerns.
I wasn't able to find anything more specific but if you used the phrase "parental hypochondria" I suspect most English speakers would understand your intent.
One potentially relevant footnote is the term Münchausen syndrome by proxy.
Münchausen syndrome by proxy — a behavior pattern in which a caregiver deliberately exaggerates, fabricates, and/or induces physical, psychological, behavioral, and/or mental health problems in those who are in their care
Münchausen syndrome by proxy is a very complicated mental illness that is somewhat related to your desired term in the sense that the illness is through "proxy"; only instead of Münchausen syndrome you are looking for something like hypochondria.
It is worth noting that you should be careful to never confuse the two. Hypochondria is usually a worry of wellbeing while Münchausen syndrome by proxy is usually an attention seeking behavior.