Gdyby chciał się oświadczyć jutro, musiałby wcześniej powiedzieć o tym jej rodzicom.

It is a hypothetical / unreal conditional:

• Gdyby chciał... = If he wanted...
• musiałby... = he would have to...

Even though the forms look past-like, the meaning can be about the present or future. Here jutro makes it clear that the situation is about the future:

• Gdyby chciał się oświadczyć jutro...
  = If he wanted to propose tomorrow...

So Polish often uses this conditional pattern for future hypotheticals, just like English uses would.

Because after gdyby, standard Polish normally uses the past-tense form without an extra -by on that same verb.

So:

• gdyby chciał = correct
• gdyby chciałby = nonstandard / not recommended

The conditional meaning is already carried by gdyby. Then the main clause usually has the conditional verb:

• Gdyby chciał..., musiałby...

That is the normal pattern.

Gdyby means if in a hypothetical or less real situation.

Compare:

• Jeśli chce, powie im.
  = If he wants, he will tell them.
  This sounds like a real possibility.

• Gdyby chciał, powiedziałby im.
  = If he wanted, he would tell them.
  This sounds hypothetical.

So in this sentence, gdyby is used because the speaker is imagining a situation, not simply stating a likely future condition.

Because in Polish, the conditional is built from a past-tense form + by, or from gdyby + past-tense form, but the meaning is not necessarily past.

Here:

• chciał does not mean wanted in a past-time sense
• it is simply part of the conditional structure

The word jutro gives the time reference:

• jutro = tomorrow

So the sentence means a future hypothetical, even though chciał looks like a past form.

Oświadczyć się is a fixed expression meaning to propose marriage.

So:

• oświadczyć się komuś = to propose to someone

The verb oświadczyć by itself usually means to declare or to state, but with się it takes on this special meaning of making a marriage proposal.

The się here is not something you should try to translate word-for-word. It is just part of the verb as it is used in this expression.

Yes, that person can be stated, but it can also be omitted if the context already makes it clear.

A fuller version could be:

• Gdyby chciał się jej oświadczyć jutro...
  = If he wanted to propose to her tomorrow...

In your sentence, that person is left out because the context probably already tells us who her is. The sentence focuses instead on the fact that he would need to tell her parents first.

Musiałby is the conditional form of musieć = to have to / must.

So:

• musi = he has to
• musiałby = he would have to

That fits the meaning perfectly:

• musiałby wcześniej powiedzieć...
  = he would have to tell... beforehand

It shows obligation in a hypothetical situation.

Because powiedzieć is perfective, and mówić is imperfective.

In this sentence, the speaker means one completed act of telling her parents. That is why powiedzieć is the natural choice.

• powiedzieć = to say / tell once, as a completed action
• mówić = to speak / to be saying / to tell in a more ongoing or repeated sense

Since he would need to do it as a single necessary step before proposing, powiedzieć fits better.

Because Polish often says:

• powiedzieć o czymś komuś = to tell someone about something

Here:

• o tym = about it

So:

• powiedzieć o tym jej rodzicom
  = tell her parents about it

Also, after the preposition o, Polish uses the locative case, which is why to becomes tym:

• to → o tym

It is dative plural.

The verb pattern here is:

• powiedzieć komu? = to tell whom?

So the people receiving the information go in the dative:

• rodzice = parents
• rodzicom = to the parents

And jej means her:

• jej rodzicom = to her parents

So the structure is:

• powiedzieć o tym jej rodzicom
  = tell her parents about it

Wcześniej means earlier or beforehand.

Here it means that the telling would need to happen before the proposal:

• propose tomorrow
• tell her parents earlier / beforehand

So the logic is:

• If he wanted to propose tomorrow, he would have to tell her parents about it first.

A close English equivalent here is beforehand.

Yes. Polish word order is fairly flexible, and changes usually affect emphasis, not the basic meaning.

For example, you could also say:

• Gdyby jutro chciał się oświadczyć, musiałby wcześniej powiedzieć o tym jej rodzicom.
• Musiałby wcześniej powiedzieć o tym jej rodzicom, gdyby chciał się oświadczyć jutro.

The original version is natural and neutral, but other orders are possible depending on what the speaker wants to stress, such as jutro, wcześniej, or jej rodzicom.