私の母は20歳のとき結婚した。

English Translation

My mother got married at the age of twenty.

Isn’t the word for 20 years old はたち?

Correct. You would read 20歳 as はたち, much as you would read 20日 as はつか. I would imagine the voice said にじゅっさい, which is also considered a correct reading! I read on a Japanese page covering the topic that, while some are hesitant to say にじゅっさい is correct, it may be used because it can flow more smoothly and could more easily dispel confusion in dense passages. As a learner, I’d consider it good practice to make use of commonplace jukujikun readings like hatachi, since the onyomi meaning can be inferred more easily if it is heard.
Ref: reading-dictionary.com[jp]

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