The example sentence in the question is fine.
Its completely expanded form is this:
- ✔ I am a writer, I am an anthropologist, and I am a teacher.
But because of its parallel structure, we are able to omit the first three words that are in common.
In the mental reinsertion of words that have been omitted from the last two items, we are even able to change the specific word a to an without a problem.
In short, the completely expanded form of the sentence can be shortened into two different ways:
I am a writer, (I am) an anthropologist, and (I am) a teacher.
→ ✔ I am a writer, an anthropologist, and a teacher.
I am a writer, (I am an) anthropologist, and (I am a) teacher.
→ ✔ I am a writer, anthropologist, and teacher.
Only the first itemized noun has to have the entirety of the subject, verb, and article in front of it.
However, if we don't omit one of those components from an itemized noun after the first, then, at least by stylistic convention, we can't omit it from any of them. Because of how parallelism works, it's an all-or-nothing affair.
In short, we normally shouldn't be writing either of the following sentences:
✘ I am a writer, an anthropologist, and teacher.
✘ I am a writer, anthropologist, and a teacher.
There are always exceptions to every rule of grammar and style, but breaking such rules should be done deliberately and with the knowledge that it's nonstandard.