Learn Hardcore Korean: toegeun hue gongwoneseo undonghaeyo. - I exercise in the park after leaving work.

퇴근 (“getting off work”) + 후 (“after”) + -에 (time particle) combine to form an adverbial phrase. So 퇴근 후에 means “after getting off work” or simply “after work.”

What's the difference between 퇴근 후 and 퇴근 후에?

퇴근 후 without -에 can act like a noun phrase (e.g. 퇴근 후 일정 “schedule after work”).
To use it as a time expression before a verb, you normally add -에 (making 퇴근 후에) to mean “after work (I do something).”
In very casual speech, some omit -에, but 퇴근 후에 is the standard adverbial form.

Can we use 퇴근 뒤에 instead of 퇴근 후에? If so, what's the nuance?

Yes. 뒤에 and 후에 both mean “after.”

뒤에 is more colloquial/spoken.
후에 is slightly more formal or common in written language.
The overall meaning remains “after getting off work.”

Why is 공원에서 used rather than 공원에?

-에서 marks the location where an action takes place (“at the park”).
-에 with a location noun either indicates a destination (“to the park”) or a static existence (“in the park”).
Since 운동해요 (“exercise”) is an action taking place there, you need 공원에서.

Could we say 공원에 운동해요? What would that imply?

No—공원에 운동해요 is ungrammatical for “exercise at the park.”

Use 공원에서 for actions.
공원에 would pair with verbs like 가다 (“go”) or 있다 (“be”), not with 운동하다.

Why isn’t there an object marker -을 after 운동?

운동하다 is a compound verb (“exercise + do”). In compounds like this, you often drop the object marker.
You could also say 운동을 해요 without changing the meaning—both 운동해요 and 운동을 해요 are correct.

What is the role of 해요 in 운동해요?

해요 is the polite present-tense form of 하다 (“to do”).
In 운동해요, 해요 attaches to 운동 (“exercise”) to form “I exercise” or “I work out” politely.

Why is there no explicit subject, and how would you add one?

Korean routinely omits the subject when it’s clear from context. Here “I” is understood.
To make it explicit, add 저는 at the front:
저는 퇴근 후에 공원에서 운동해요.
(“As for me, after work I exercise at the park.”)

How would you express “I go to the park to exercise after work” if you want to emphasize the going action?

Use the purpose construction -러 가요 (“go in order to …”):
퇴근 후에 공원에 운동하러 가요.
Here, 공원에 is the destination (“to the park”) and 운동하러 가요 means “go to exercise.”