Why are feet and teeth the only body parts for which specialists don't have to go to med school?

orthopedists can deal with feet, But the separation of medicine and dentistry goes back a thousand or more years and has been that way so long its in cement at this point. Same with the medical specialties and the surgical ones.  Henry the VIII established the College of Medicine and Surgery as a linked entity. You could look up the history of podiatry,

You really need to do some research on the subject. You are so wrong on everything.

BTW every bodypart has a medical profession associated with it and they all have to get educated and they all get a degree. Some only require a B.S. Some a M.S. and many require that PhD. Depends on requirements and where your interest lay.

Podiatrist and dentist DO GO TO SCHOOL AD HAVE DEGREES. And yes they do have letters after their names. Same with Chiropractors. They have a lot of schooling as well.

Dentist and orthodontists both have years of schooling. Orthodontists are more specialized and do more surgeries. Dentist is really no different than going to your General Practitioner that you go to for your regular checkup and your ailments.

In the United States, medical and surgical care of the foot and ankle is mainly provided by two groups of physicians: podiatric surgeons (who hold the degree of Doctor of Podiatric Medicine or DPM) and orthopedic surgeons (MD or DO).[36][citation needed]In the US Podiatrists:The first two years of podiatric medical school is equivalent to training that M.D. and D.O. students receive, but with an emphasized scope on foot, ankle, and lower extremity. Being classified as a second entry degree, in order to be considered for admission an applicant must first complete a minimum of 90 semester hours at the university level or more commonly, complete a bachelor's degree with emphasis on general/organic chemistry, biochemistry, biology, etc. In addition, potential students are required to take the Medical College Admission Test (MCAT). The DPM degree itself takes a minimum of four years to complete.

I could go on. But i want to call you out and you can do more finger walking to get the real answers.

They do go to school. In the U.K. a Dentist has 5 years training. It's about 3 years for feet.

They do go to med school 

Optometrists go to optometry school, a four year post graduate program. There are medical doctors who specialize in eyes as well. I think that the education for podiatrists and dentists is similar to optometrist. I think that chiropractors go to school about as long as well. There are nurse practitioners and physician assistants who  treat a broad variety of diseases often with no real supervision.

Dentistry is unique in that there is no real specialty for medical doctors to treat teeth.