Learn Hardcore Spanish (Latin America): Mis padres prefieren hablar conmigo en lugar de usar castigos fuertes. - My parents prefer to talk with me instead of using harsh punishments.

You only conjugate the first verb (prefieren).
Using prefieren hablan would be like saying “they prefer they talk,” which is ungrammatical in Spanish in this structure.

Why is it conmigo and not con mí?

Spanish has special combined forms for con + mí / ti:

con mí → conmigo (with me)
con ti → contigo (with you, singular informal)

So hablar conmigo literally is to talk with me.
Saying con mí is incorrect in standard Spanish.

Could I say a mí instead of conmigo, like hablar a mí?

No, not in this sentence.

hablar con alguien = to talk with someone (a conversation)
hablar a alguien = to speak to someone (more like talking at someone, or addressing them)

In the context of parents preferring conversation over punishment, Spanish naturally uses:

hablar conmigo = talk with me

Hablar a mí sounds unnatural here and would not be used.

What is the difference between mis padres and something like mis papás?

Both are common in Latin America, but there is a nuance:

mis padres = my parents (more neutral / slightly more formal)
mis papás = my mom and dad (more informal, affectionate)

In this sentence, either mis padres or mis papás would work in Latin American Spanish, depending on the tone you want. Mis padres sounds a bit more neutral or textbook-like.

Why is there no article before castigos fuertes? Why not los castigos fuertes?

Spanish often omits the article when talking about things in general, especially after verbs like usar (to use):

usar castigos fuertes = to use strong punishments (in general)
usar los castigos fuertes would sound like “to use the strong punishments,” referring to specific punishments already known in context.

Here, the meaning is general (any strong punishments), so no article (no los) is the most natural choice.

What’s the role of en lugar de here? Is it the same as en vez de?

En lugar de means instead of. It introduces an alternative action:

prefieren hablar conmigo en lugar de usar castigos fuertes
= they prefer to talk with me instead of using strong punishments.

En vez de is very similar and also common in Latin America:

en lugar de usar castigos fuertes
en vez de usar castigos fuertes

Both are correct; en lugar de is just a bit more neutral/formal. The structure is the same: en lugar de + infinitive / en vez de + infinitive.

Why is it usar castigos fuertes and not dar castigos fuertes?

Both exist, but they are used differently:

usar castigos fuertes = to use strong punishments (focus on the method/strategy)
dar un castigo fuerte = to give a strong punishment (a specific act)

In this sentence, we are contrasting methods (talking vs. using strong punishments), so usar is a natural choice.

You could say:

…en lugar de darme castigos fuertes = instead of giving me strong punishments

But then you’d usually add a pronoun (darme, give me). The original sentence sounds more general and method-focused.

Why is castigos plural? Could it be un castigo fuerte?

Both are possible, but they give slightly different nuances:

usar castigos fuertes (plural)
= using strong punishments in general, as a type of discipline.
usar un castigo fuerte (singular)
= using a strong punishment (one punishment in some context).

In the sentence, the idea is about their overall approach to discipline, so the plural castigos fuertes works better to describe a general practice.

Why is fuertes placed after castigos? Could it be fuertes castigos?

The most typical order is noun + adjective:

castigos fuertes = strong punishments

You can put some adjectives before the noun, but that often adds emotion, emphasis, or changes the nuance. Fuertes castigos is possible in very formal, literary, or dramatic language, but in everyday Latin American Spanish, castigos fuertes is more natural and neutral.

So for regular speech, keep castigos fuertes.

Why is de used after en lugar? Could I say en lugar usar?

The fixed expression is en lugar de:

en lugar de + infinitive / noun
- en lugar de usar castigos fuertes
- en lugar de castigos fuertes

Leaving out de (en lugar usar) is incorrect. Think of en lugar de as a single chunk meaning instead of.

How do I know to use prefieren and not something like les gusta here?

Both preferir and gustar express liking, but they behave differently:

preferir works like a regular verb:
- Mis padres prefieren hablar… = My parents prefer to talk…
gustar works backward (literally “to be pleasing to”):
- A mis padres les gusta hablar… = Talking is pleasing to my parents.

Preferir is stronger and specifically expresses preference between options (A instead of B). Since the sentence contrasts talking vs using punishments, prefieren is the best choice:

Mis padres prefieren hablar conmigo en lugar de usar castigos fuertes.

Could I say Mis padres prefieren que hablen conmigo instead of prefieren hablar conmigo?

That would change the meaning and be ungrammatical as written:

prefieren hablar conmigo
= they prefer to talk with me (they themselves do the talking)
prefieren que hable conmigo
= they prefer that (someone) talk with me

If you use que + subjunctive, the subject of the second action is usually different from the subject of the first. For example:

Mis padres prefieren que el terapeuta hable conmigo.
= My parents prefer that the therapist talk with me.

In your original idea, prefieren que hablen conmigo, we don’t know who “they” are, and it’s awkward. For “they prefer to talk with me,” the infinitive hablar is the correct and natural choice.