I know there's another question about "se", but I don't understand the answer or know which of the scenarios described refers to this one.
While learning Spanish, I'm supposed to know the sentence
Mis padres se llaman __ y __.
I understand the word "llaman" is "call". So, I translate "mis padres llaman __ y __" as "my parents are called _ and _". What does "se" do?
Here is what I understand from my English speaking point of view.
In English, we say
"My parents are called X and Y".
Who called them? The statement didn't say who, but that's how we use passive tense. So in Spanish, that is the purpose of "se".
Without the "se" it would sound as if you are saying: "My parents call...(who/what)"? You are normally expected to hear who are they calling or what are they calling.
"Mis padres llaman..."
"Mis padres llaman a su amigo Benny."
"My parents call their friend Benny."
Therefore, if you say
"Mis padres llaman a John y Jane".
It would actually mean My parents call John and Jane. My parents are simply calling someone else named John and Jane for something, maybe for a dinner.
Yo me llamo Diego. Mis padres (ellos) se llaman __ y __
Se is just the third person pronoun, like me is the first person pronoun.
I don't know what is your mother tongue, but since you seem fluent in at least these two (English and Spanish), you are already aware that we say in English "what is your name" while in Spanish we ask "Cómo te llamas?", and not "Qué es tu nombre" which would be the literal transalation.
Llamarse is a pronominal verb, not a reflexive. I have a name, but when I say "Me llamo Diego" I don't mean I call myself nor that I give myself that name. It means "I'm known by this name".
A reflexive verb is that in which who performs the action is also the recipient of the action. A pronominal verb is a special case in which the verbs must be conjugated with a pronoun. Explaining reflexive and/or pronominal verb might be out of the scope of this question, but please, feel free to ask other questions about it if you need to, or check learning resources about the topic.
Basically we use that grammatical construction in Spanish to say "my/your/his name is" and that construction has a reflexive pronoun. Se is the third person pronoun, which is what you should use for "ellos".
It's an important question and a good question. Here's one use of llamar:
At 8:00, I call my cat to come in and eat.
A las 8:00, yo llamo a mi gato a venir a comer.
Here's another:
Sometimes I call my cat "Crazy" because of his high jumps.
A veces yo llamo a mi gato "Locochón" debido a sus grandes saltos.
But that's just a silly nickname. His name is actually Miguelito. To express that, I'll use llamar with a special pronoun tucked before it:
Mi gato se llama Miguelito.
A literal, word for word translation of this would be
My cat calls himself Miguelito.
But a correct translation of this expression would be
My cat's name is Miguelito.
Here's what it looks like, in the other conjugations:
Yo me llamo Samuel.
¿Tú te llamas Samuel?
Nosotros nos llamamos la Familia Pérez.
Vosotros (pardon me if I skip this -- we don't use this in Mexico and I believe it might vary from one country to another, I don't want to steer you wrong -- but someone else is welcome to edit my answer and fix this up)
Mis padres se llaman Conchita y José Manuel.
You could make a word for word, literal translation from English and say:
My parents' names are Daisy and Pablo.
Los nombres de mis padres son Daisy y Pablo.
and people would understand you; but it would sound a bit awkward.
