Learn Hardcore Japanese: ryousin to watasi ha haru ni bizyutukan de atarasii e wo mimasu. - My parents and I will see new paintings at the art museum in spring.

In Japanese, は marks the topic of the sentence, and it attaches to the end of the entire noun phrase you want to treat as your topic. Here, you want both “my parents” and “I” to be the topic together, so you join them with と (“and”) and then attach は:
両親と私は…

If you wrote 両親は私と…, you’d be making “my parents” the topic and then mentioning “me and them” separately, which changes the nuance.

Is it okay to say 私と両親は instead of 両親と私は?

Yes, it’s grammatically correct, but in Japanese (especially polite contexts) it’s customary to mention others before yourself. So 両親と私は sounds more natural and humble than 私と両親は.

Why is に used after 春 in 春に?

The particle に marks a specific point in time. To say “in spring” you use 春に. Without に, 春 would sound like a general topic or lack a clear time indication.

What does で do in 美術館で?

The particle で indicates the place where an action occurs. Here, 美術館で tells us that the action (見ます – “will see”) happens at the art museum.

Why do we use を after 新しい絵 in 新しい絵を見ます?

The particle を marks the direct object of a verb. It shows that 新しい絵 (“new painting/paintings”) is what you will see with the verb 見ます.

Why does 新しい come before 絵? Do all Japanese adjectives work that way?

Yes. In Japanese, both い-adjectives and な-adjectives directly precede the noun they modify. Since 新しい is an い-adjective, it naturally goes before 絵 to form 新しい絵.

How can I tell if 絵 is singular or plural?

Japanese nouns don’t change form for number. 絵 can mean “a painting” or “paintings” depending on context. If you need to emphasize plurality, you might add words like 何枚かの絵 (“several paintings”), but often it’s clear from the situation.

Why isn’t there a subject marker が in this sentence?

Because the sentence already uses は to mark the topic, which also functions as the subject here. You only use が when you need to introduce a new subject or emphasize it. Since 両親と私 is already your topic, が isn’t needed.

Could I say 両親と私は春に美術館に行きます instead?

Yes—if you want to say “we will go to the museum in spring.” Note two changes:

Use the verb 行きます (“go”) instead of 見ます (“see”).
Change the location particle from で to に: 美術館に行きます means “go to the museum.”