
nebeneinander
side by side
Spatially directly at the side of something or someone; side by side.
Im Regal stehen die deutschen und die englischen Bücher nebeneinander.
On the shelf, the German and English books are standing side by side.
next to each other / side by side
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side by side
Spatially directly at the side of something or someone; side by side.
Im Regal stehen die deutschen und die englischen Bücher nebeneinander.
On the shelf, the German and English books are standing side by side.

in parallel
Taking place at the same time or in parallel.
Es ist schwierig, wenn zwei so wichtige Projekte nebeneinander laufen.
It is difficult when two such important projects run in parallel.

to live side by side
To be in a relationship in which the partners live together without emotional closeness or shared activities.
Sie haben sich nicht getrennt, aber sie lebten die letzten Jahre nur noch nebeneinander her.
They did not separate, but in recent years they only lived side by side.
'Nebeneinander' (next to each other/side by side) is a compound made up of the preposition 'neben' (next to) and the reciprocal pronoun 'einander' (each other). So it literally means 'next to the other one'.
Die beiden Kinder saßen still nebeneinander auf der Bank.
The two children sat quietly side by side on the bench.
As an adverb, 'nebeneinander' can appear in different positions in the sentence. It can be in position 1 to emphasize the arrangement, or after the conjugated verb.
Nebeneinander wirkten die beiden Häuser sehr unterschiedlich.
Side by side, the two houses looked very different.
When 'nebeneinander' is used with abstract nouns like 'Projekte' (projects), 'Aufgaben' (tasks), or 'Systeme' (systems), it usually means 'at the same time' or 'in parallel'.
In unserem Unternehmen existieren alte und neue Softwaresysteme nebeneinander.
In our company, old and new software systems exist in parallel.
The phrase 'nebeneinander her leben' (to live side by side) is a fixed expression. It describes couples or partners who live together but no longer have an emotional connection or shared interests.
Nach zwanzig Jahren Ehe lebten sie nur noch nebeneinander her.
After twenty years of marriage, they only lived side by side.
parallel
Parallel and nebeneinander can indicate that several processes are taking place at the same time. Parallel is the usual neutral term for processes running simultaneously; nebeneinander places more emphasis on their existing side by side. Parallel often fits technical or organizational descriptions, while nebeneinander is suitable when simply describing simultaneous coexistence.

Während der Übergangszeit bestehen das alte und das neue Verfahren nebeneinander.
During the transition period, the old and new procedures coexist side by side.
Laut Projektplan laufen das alte und das neue Verfahren parallel.
According to the project plan, the old and new procedures are running in parallel.