halacha - Are Shiros considered one parshah or several?

The two Shiros, Shiras HaYam and Shiras Ha’azinu, are both written slightly differently from one another, but are both written in a “brickwork” format, rather than the typical paragraph format (YD 275:3-5). From observing Sifrei Torah, the gist of it seems to be a layering of text, wherein the words are broken up into several word phrases, which are then layered line by line, with gaps in the middle.

What are considered the parshah breaks for these? Both songs are preceded by and followed by a Pesuchah - are those the bounds of the parshah, and one could start an aliyah in the fourth passuk of either song? Or are the breaks in the middle of the lines considered a new parshah each time?

When a ba’al korei reads a parshah, he may not begin an aliyah within three pesukim of the beginning of a parsha, nor may he end one within three pesukim of the end of a parsha (OC 138). To that end, I am aware that the practice is to have several parshah breaks during Shiras Ha’azinu, but none in Shiras HaYam (at least among Ashkenazim). Does this mean that the Shiros are a single parshah? If so, is this limited to Ha’azinu, but the breaks in Beshalach are Parshah breaks? Or does this in theory apply to Beshalach as well, just that in practice it wouldn’t occur without adding aliyos according to our minhag?

In the event that the gaps during a Shirah are not considered Parshah breaks, I would like an explanation as to why that is the case. In the event that Shiras HaYam and Shiras Ha’azinu are different, I would also like an explanation of that as well.