Don't die on this hill, OP. You need the childcare and your parents, while invasive and incredibly annoying, aren't actually doing any harm to your children! They're harming YOU and your husband, that's for sure, by constantly undermining you and getting in the way. But right now is the time to swallow your pride, smile and let them take the lead.
A word about undereye circles. They can come from sleep deprivation but also from allergies. My son is allergic to all kinds of things, including cats, pollen and nuts, and has always had undereye circles.
Learn to not get upset when your parents try to micromanage MINOR details. This is the example of a minor detail. The doctor may reassure them, or uncover various minor ailments.
If you can persuade your parents to get treatment for anxiety, that would be great! I won't hold my breath, though. They sound like die-hard, life-long, anxious people who are in denial about their anxiety... just like my parents

(I crossed the pond to get away from them, FYI!).