Yes, it is possible to parent without medicines and drugs. There are some children who may in fact be medicine dependent (maybe less than 5%), the rest of the OTC (over the counter) and prescription medications are by and large unneeded and even harmful.
The classic medicinal 'chase' is for ear infections. Antibiotics are used, that actually lower the body's immune system and weaken the person's ability to fight the disease on its own. Often, this cycle of infection-antibiotic-weakened immune system is repeated time after time with multiple side effects.
Another classic is chasing away a fever (which is usually there helping the body fight off an infection of some kind) with a tylenol, aspirin, or some other anti-pyretic, (anti-fever). This lowers the temperature of the body in direct opposition to what the body is actually attempting to do!
Forgotten is the axiom, 'Let the body heal itself.' We are a self-healing magnificent-miracle making machine. Our body is working to heal itself 24-7. It needs some decent nutrition, cardio-vascular exercise, sleep, hydration and then it can do most of the rest on its own...it simply needs no interference.
Interference can come in many forms: physical, chemical, emotional, spiritual, etc. Physical can include traumas or injuries, and also subluxations or misalignments in the spine (the work of we chiropractors). Chemical can be medicines, bad air, bad water, bad nutrition. Emotional and spiritual are self-explanatory...
Trust the body not the OTC crud or the pharmacy. (Did you ever think that it's really not in pharmaceutical industry's best interests to get people well?). Don't rush to the medicine cabinet (or better yet, have one filled only with first aid stuff and homeopathic remedies), or to the ER.
You and your children will be glad that you did.