A Dentist With Bad Teeth

[CaseyA]

A dentist with bad teeth is to a hairdresser with bad hair.

Is this sentence grammatical?

[Raymott]

Is this sentence grammatical?

No.
1. Bad teeth to a dentist is as bad hair to a hairdresser.
2. Bad teeth is to a dentist as bad hair is to a hairdresser.
3. A dentist with bad teeth is as a hairdresser with bad hair.
4. A dentist with bad teeth is similar/analogous to a hairdresser with bad hair.

You can't just say, "A is to B". A proper sentence is "A is to B as C is to D" - ie. 2.