...I'll bet the farm that no English teacher... told her that pronouns like "it," "they," "them," and "their" were not allowed.
Agreed. 100%.
The teacher may have put restrictions on them, such as the pronouns should match their antecedents in number, but "not allowed"? That's the part which I suggest is disremembered.
Alternatively, the sentence in question in the OP's essay might have had two plural nouns preceding the use of the pronoun (for example, perhaps 'the stableboys' and 'the trainers'), thus making it unclear (syntactically) to which of the plural nouns 'they' or 'their' referred.
In formal writing, which an essay for English class is, it must be clear
syntactically (not just from the context) for which noun the pronoun is standing in.
For example, let's take the following example.
Clem and Bill went into the stable. Clem picked up a halter while Bill closed the door, and he put it on the old grey mare.
From the
context, it's clear that 'he' refers to Clem, not Bill. But from a purely
syntax point of view, it's not clear, and, therefore, for formal writing, the pronoun would be incorrect usage. For that matter, the pronoun 'it' would be similarly
syntactically unclear, since it could refer to either the halter or the door.
Since it's the job of English teachers to teach formal writing skills to students, I can imagine that the above sentence in an essay would be marked down.
And the fix is, of course, quite easy. Either of the following examples works, as would a number of others.
Clem and Bill went into the stable. Clem picked up a halter and put it on the old grey mare, while Bill closed the door.
or
Clem and Bill went into the stable, and Bill closed the door. Clem picked up a halter and put it on the old grey mare.
I'd bet dollars to donuts that this was what the teacher was trying to convey about pronouns, and that the OP didn't understand the point or misremembers it.