I originally learned to play go using Japanese-style territory counting to score the game.

For a long time, I thought that parallel openings favoured black more than cross openings, because cross openings lead to games with lower territory counts, and so komi would have a bigger effect on the score for white. It occurred to me recently though, that using area scoring, the effect of territory vs dame is somewhat lost, and yet komi is effectively the same. So, perhaps this idea is just an illusion of the scoring method by which I learned to play, and doesn't actually have any effect on the game.

Given that, does one style of opening actually lead to a more favourable game for black or white?

I know looking at Leelazero and Katago that they seem to favour parallel openings, but when you look at the percentages the difference is small enough that it could just be noise. I gather that historically parallel openings have been favoured by black as well, but now my earlier reasoning for it seems wrong.

Do any of you have a better sense of this than I do, or more information on why pros would prefer parallel openings?

I think it is largely what you explained. To put it differently: Black wants to raise the value of having the first move. It is logical enough that a parallel opening creates opportunities to play on a larger scale, using the stones on the board to create high value areas where black gets the first move.

However, a more objective analysis might be contrary to this "logical" conclusion. At least it seems to be the case that it is very hard to achieve an advantageous position in the early opening, even when considering human tangibles like fast development and initiative. The computer's assessment appears to be that it is not only possible but easy to keep up with black with the right komi.

My personal experience is that more dynamic play in the early opening leads to more chances (whichever side you are playing) and also importantly it avoids constantly repeating too similar openings. That is you encounter new situations if you try to achieve an advantage and this is good material to study, but it could be almost useless from an objective standpoint because the computer says it is all very much the same.

I suppose it could also be an artifact of the selection process for a cross opening. Both players have to agree to play a cross opening in some sense, so if you were to just assign an equal probability for either player deciding to aim for a cross opening, you would only get one 25% of the time.

I don't have the time to look it up with some sort of database evaluation right now, but I'd be curious what the actual percentage of cross vs parallel vs weird openings is in professional play now and for different time periods.

It is more usual for pros allow a cross fuseki than not but it is still not so common to accept it. I don't have very good statistics, it will also depend on the exact corner moves, but white playing in the adjacent corner is very common in pro play. On the other hand it seems to be much much more common in the typical game on online Go servers for white to play in the opposite corner (even when black plays a komoku). I have always found this difference between the professional and amateur world odd.

Basically, it is more "normal" to play the second move in the adjacent corner than it is not. The statistic argument probably doesn't explain the choice of playing parallel fuseki.

The traditional thinking about cross (tasuki) openings was that they led to early fighting, which (especially in no-komi games) Black might do best to avoid. At the very least it changed the joseki palette, because the ladder breakers changed.

But that thinking applied to pro games. I very much doubt that it makes a ha'porth of difference in amateur games.

Is this a view that was ever revised in the komi era?

Revised but not abandoned, especially when long ladder-prone josekis such as the Avalanche and Magic Sword were being explored.

I suppose the question then is whether komi is high enough now that black would want to play an active game with early fighting to make up for it.

There could be a confirmation bias at play. If the governing idea is that parallel openings lead to cooperating stones hence to moyos or otherwise a leverage of the first move advantage, then players may explore it more and find ways to create that advantage, while also exploring how to reduce that advantage. So parallel openings are played and studied more in an ever growing gambit, without checking the original assumption.

Or it's a psychological fight: White 2 can force a parallel opening but doesn't, which suggests she favors diagonal openings, so Black 3 plays parallel.

I'm speculating.