der Umzug
move
The change of residence from one apartment or house to another.
Der Umzug in die neue Wohnung war anstrengender als gedacht.
The move into the new apartment was more exhausting than expected.

move / parade
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move
The change of residence from one apartment or house to another.
Der Umzug in die neue Wohnung war anstrengender als gedacht.
The move into the new apartment was more exhausting than expected.

'Der Umzug' (move/parade) is the nominalization of the separable verb 'umziehen' (to move / to change clothes). It describes the action of the verb as a noun. This helps you connect the two main meanings (moving house and parade) with the respective verbs.
Wir ziehen nächste Woche um, der Umzug wird sicher stressig.
We are moving next week; the move will certainly be stressful.
You can easily recognize the meaning from context. If it is about 'Wohnung' (apartment), 'Möbel' (furniture), or 'Kisten' (boxes), it means moving house. With words like 'Karneval' (Carnival), 'Fest' (festival), or 'Musik' (music), it refers to a parade.
Für den Umzug müssen wir noch viele Kisten packen.
We still have to pack many boxes for the move.
When it is about moving house, typical verbs are 'planen' (to plan), 'organisieren' (to organize), or 'bevorstehen' (to be imminent). A common phrase is also 'beim Umzug helfen' (to help with the move).
Unser Umzug steht kurz bevor.
Our move is coming up very soon.
In the context of a parade, 'Umzug' is often combined with festive occasions, e.g. 'der Karnevalsumzug' (the Carnival parade) or 'der Festumzug' (the festive parade). Typical verbs are 'sich ansehen' (to watch) or 'teilnehmen' (to participate).
Tausende Menschen haben sich den traditionellen Festumzug angesehen.
Thousands of people watched the traditional festive parade.
train / move / draft
'der Zug' and 'der Umzug' can both mean a group of people moving through the streets in an organized way. In this sense, 'Zug' is more general, for example in a demonstration or parade. 'Umzug' often means a festive parade, especially at Carnival or a town festival. They can be confused because 'Umzug' contains the word 'Zug'. But be careful: 'der Zug' very often means the rail vehicle, and 'der Umzug' very often means moving to a different home. In these meanings, they are not the same.

Beim Karneval sehen viele Familien den bunten Umzug an.
At Carnival, many families watch the colorful parade.
Der Zug der Demonstranten zog langsam durch die Innenstadt.
The procession of demonstrators moved slowly through the city center.