There are a number of things that need to be corrected before answer.
You example sentence ought to be
Mis padres (me) exigen que lea tres libros cada día.
The reality is, it's virtually always optional from a grammatical standpoint.1 It's useful when there could be confusion in the subordinate clause:
- Mis padres exigen que lean tres libros cada día.
Are they demanding this of y'all (formal)? Or of some other group of people? Or even of themselves?
- Mis padres exigen que lea tres libros cada día.
Are they demanding this of me? Or someone else?
There are two ways to clarify. You can either explicitly put the subject in the subordinate clause, or you can use an indirect object pronoun (this will always be an indirect object pronoun, because the subordinate clause is the direct object, it's just a noun clause rather than a pronoun):
- Mis padres le exigen que lea tres libros cada día.
Using indirect object: They're demanding, just thankfully it's not me.
Mis padres me exigen que lea tres libros cada día.
Using indirect object: They're demanding of me!
Mis padres exigen que ella lea tres libros cada día.
Using explicit subject: They're demanding of her.
- Mis padres exigen que yo lea tres libros cada día.
Using explicit subject: They're demanding of me!
1. The only time that you would have to use the indirect object pronoun is if person A is telling person B to have person C do something. Like, say, I dunno, my boss's boss tells him to tell me to do something: «La jefa le exigió (al otro jefe) que haga yo algo». But even then, if you make the indirect object explicit, the indirect object pronoun becomes optional again: «La jefa exigió al otro jefe que haga yo algo» is perfectly fine.