To add, the number of days absence that can trigger a fine is currently up to the LA, so it varies depending on where you are. There is at least one that will fine for a single day's unauthorised absence. Most won't fine for less than 5 days absence unless there have been other unauthorised absences.
I can understand why many parents think schools aren't allowed to authorise absence. Many heads promote the myth that they have no discretion, and that there are rules they must follow. The law, however, states that absence can be authorised if, "the proprietor, or a person authorised by the proprietor... considers that leave of absence should be granted due to the exceptional circumstances relating to that application." That's it. The regulations do not contain any definitive list as to what qualifies as exceptional circumstances, nor has the government issued any such list. The proprietor for a state school is the governors, so the head is the person authorised by the proprietor. It is therefore entirely up to the head to decide what qualifies as exceptional circumstances and will therefore lead to absence being authorised.
Some heads take a hard line on this and refuse to authorise any absence. A few take a very relaxed approach and authorise almost anything, although they risk the wrath of Ofsted if the attendance figures are poor as a result. Many look at each case individually, as the regulations intend. There is, I believe, guidance,
from some unions on what should be regarded as exceptional circumstances, and some websites aimed at teachers also offer guidance, but this guidance is not official and is not in any way binding.