With a DDFSTRT / STOP of $30 / $d8, you get 16px of horizontal overscan on the right, and can use the scroll register. (Not all parallax scrolling requires the scroll register.)
Maximum horizontal overscan is 384, but there are some monitor and gameplay limitations to consider:
You might be wasting time on rendering graphics many don't see, perhaps enemies or bullets that you'd very much like to see.
If you use a 5:4 or 4:3 flatscreen monitor for display, horizontal overscan will likely be stripped.
The game's programmer must also know a little bit about the Copper. Most OCS games did very little with the Copper, and perhaps they were conservative, or it made the port quicker. Maybe it's as rare as
Amiga-First games, but probably it's rarer. The reason being: it's not just the play area that should be bigger. Panels etc. graphics must also be drawn for it, and you would have noticed quite clearly if e.g. the score or number of lives were cut off by your TV (which was the most common display during the OCS game explosion).
Sprites:
As you move the left border farther out, more and more sprites are lost. This is by design: you are asking DMA slots near the left border to transfer bitplane data instead of sprite data. At the maximum overscan the first sprite is still displaying, also by design: to still be able to point and click in Workbench.