"Suzume" is the kun-reading for that kanji, while "zaku" sounds like an on-reading, but my dictionary only lists "jaku", "jan", "saku", and "shaku". If you haven't delved into kanji study yet, "kun-reading" (訓読み / kunyomi) refers to the Japanese-originating pronunciation for the character, while the "on-reading" (音読み / onyomi) refers to Chinese-derived pronunciations. Kunyomi are typically used in verbs, i-adjectives, and single-character nouns, while onyomi are typically used in 熟語 / jukugo (compound words). That's just a rough guide though, and there are plenty of exceptions.

The word for "sparrow" is 雀, pronounced すずめ, as it's a single character noun and so uses the kunyomi. None of the jukugo I can find for 雀 are very useful, so I wouldn't worry too much about learning the onyomi for this one.

Edit: By "suzaku", do you mean this word: 朱雀? As far as I can tell, this is a really obscure term referring to a constellation. You'll find that many such words don't often follow normal pronunciation conventions. And, as it so happens, the onyomi for 朱 is シュ, and not ス, so neither of the kanji in that word are using their typical readings.