
nebenbei
on the side
In addition to a main activity; as a side activity.
Er studiert Vollzeit und arbeitet nebenbei in einem Café.
He studies full-time and works on the side in a café.
on the side / incidentally
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on the side
In addition to a main activity; as a side activity.
Er studiert Vollzeit und arbeitet nebenbei in einem Café.
He studies full-time and works on the side in a café.

incidentally
Mentioned in passing, as information that is not very important.
Ach, nebenbei bemerkt, dein Paket ist heute angekommen.
Oh, incidentally, your package arrived today.

at the same time
Without much effort or concentration, almost as if by itself.
Beim Kochen hört sie nebenbei einen Podcast.
While cooking, she listens to a podcast at the same time.
'Nebenbei' is made up of the prepositions 'neben' (beside/next to) and 'bei' (by/at/with). Literally, it means 'alongside it' or 'in addition to it' and describes something that happens in parallel with another thing.
Sie lernt für ihre Prüfung und hört nebenbei Musik.
She studies for her exam and listens to music at the same time.
As an adverb, 'nebenbei' can stand flexibly in the sentence. If it stands in position 1, the conjugated verb follows immediately (inversion).
Er hat das nebenbei erledigt.
He took care of that on the side.
In this meaning, 'nebenbei' is often connected with activities that bring in extra money or knowledge. Typical phrases are 'nebenbei arbeiten' (to work on the side), 'nebenbei Geld verdienen' (to earn money on the side), or 'nebenbei eine Sprache lernen' (to learn a language on the side).
Viele Studenten müssen nebenbei arbeiten, um ihr Studium zu finanzieren.
Many students have to work on the side to finance their studies.
The expression 'nebenbei bemerkt' (incidentally/by the way) is a very common way to weave additional, less important information into a conversation.
Das Meeting war sehr produktiv. Nebenbei bemerkt, es gab auch sehr guten Kaffee.
The meeting was very productive. Incidentally, there was also very good coffee.
by the way
Both words can casually introduce additional information. „Übrigens“ usually marks a new thought or topic of conversation. „Nebenbei“ emphasizes more strongly that the information is mentioned only in passing and is not particularly important.

Nur nebenbei erwähne ich, dass der Kurs morgen eine Stunde später beginnt.
I mention only in passing that the course starts an hour later tomorrow.
Übrigens beginnt der Kurs morgen eine Stunde später.
By the way, the course starts an hour later tomorrow.