Learn Russian: Эти хорошие люди — мои родители.

Because they are describing plural nouns (люди and родители). In the nominative plural, the demonstrative pronoun этот (this) becomes эти (these), and the possessive pronoun мой (my) becomes мои.

Why does the plural adjective хорошие end in -ие instead of the usual -ые?

Normally, plural adjectives end in -ые (like новые for 'new'). However, хорошие has the consonant ш. According to the Russian 7-letter spelling rule, you can never write the letter ы after г, к, х, ж, ч, ш, щ. Instead, you must use и, making the ending -ие.

Wait, isn't the word for 'person' человек? Why is 'people' люди?

Yes, человек means 'person'. However, in Russian, the plural of 'person' is a completely different word. When you want to say 'people' as a plural subject, you must use the word люди.

Why is there a dash in the middle of this sentence?

In Russian, the present tense of the verb 'to be' (am, is, are) is almost never spoken or written. When a sentence links a noun subject to a noun definition (like 'people are parents'), a dash is often used in writing to stand in for the missing 'are'.