Learn Hardcore Chinese: jiějie bìyè nà tiān, fùmǔ jì gāoxìng yòu yǒudiǎnr nánguò. - On the day my sister graduated, our parents were both happy and a bit sad.

In Chinese, time expressions can directly modify the sentence without a preposition.
姐姐毕业那天 literally means “the day (when) my older sister graduated” and already answers “when?” for the main clause, so 在 is not required.
You can say 在姐姐毕业那天, but it sounds a bit heavier and is less common in everyday speech.

Why is there no 的 in 姐姐毕业那天? Could I say 姐姐毕业的那天?

Both 姐姐毕业那天 and 姐姐毕业的那天 are grammatically correct.

姐姐毕业那天 is a bit more compact and colloquial.
姐姐毕业的那天 sounds slightly more explicit and emphasized, like “that day when my sister graduated.”

In many cases (especially with time words like 天, 时候), the 的 can be omitted without changing the meaning much.

What exactly does 既…又… mean, and how is it used?

既…又… links two parallel, coexisting qualities, like “both … and …” in English.
In the sentence, 既高兴又有点儿难过 means “(they were) both happy and a bit sad.”

Basic pattern:

Subject + 既 + Adjective/Verb Phrase 1 + 又 + Adjective/Verb Phrase 2

You can use it with adjectives (既大又漂亮) or verbs (既工作又学习).

Can 既…又… be replaced with 又…又… or 又…也… here?

Yes, you could say 父母又高兴又有点儿难过, and it would be natural in conversation.
既…又… often sounds a bit more formal or written, while 又…又… is very common in spoken Chinese.

又…也… (like 又高兴也有点儿难过) is understandable but less standard; the usual pairs are:

既…又…
又…又…
又…还… (e.g. 又高兴还很紧张)

What is the nuance of 有点儿 in 有点儿难过?

有点儿 before an adjective means “a little / somewhat / rather,” and often carries a slightly negative or undesired feeling.

有点儿难过 = “a bit sad / somewhat upset” (not a good thing)

With positive adjectives, 有点儿 is used less often and may sound like the positive thing is too much (e.g. 有点儿贵, 有点儿吵).
If you just want a neutral “a little” with a noun or after verbs, you usually use 一点儿 instead.

Could we say 一点儿难过 instead of 有点儿难过?

一点儿难过 is not wrong, but it is less natural as a straight description of feeling.

有点儿难过 = “feel a bit sad” (very common, natural)
难过一点儿 or 有一点儿难过 can work in some contexts, but 有点儿难过 is the most idiomatic choice for “a little sad” here.

So in this sentence, 有点儿 is the best fit.

Does 难过 here mean “sad” or “to have a hard time”? Is it an adjective or a verb?

In this sentence, 难过 functions as an adjective meaning “sad / feeling bad emotionally.”
Chinese adjectives often behave like stative verbs, so 难过 can also work as “to feel sad / to feel bad.”

Here, 有点儿难过 is “(they) were a bit sad,” combining degree (有点儿) + adjective (难过).

Why is it just 姐姐, not 我的姐姐 (“my older sister”)?

Chinese often omits possessives with close family members when the context is clear.
If you are talking about your own family, 妈妈, 爸爸, 姐姐, 弟弟, etc. are usually understood as “my mom,” “my dad,” “my older sister,” and so on.

You can say 我的姐姐毕业那天, but it usually sounds a bit more formal or explanatory, for example if you have to distinguish your sister from someone else’s.

What exactly does 父母 mean? Is it singular or plural?

父母 literally means “father and mother” and is treated as a pair, similar to “parents” in English.
Grammatically it behaves like a plural noun, but Chinese does not change the noun form for singular/plural.

If you need to emphasize “my parents,” you can say 我父母 or 我的父母; here it is clear from context that they are their parents.

Is 既高兴又有点儿难过 describing the parents as having mixed feelings at the same time?

Yes. 既…又… in this sentence expresses that the parents simultaneously felt two contrasting emotions:

高兴 – happy (that the sister graduated)
有点儿难过 – a bit sad (perhaps because she is grown up, leaving home, etc.)

So the phrase neatly captures the idea of “mixed feelings” happening at the same time.