How do you reckon it is "dominated by right-wing parties"? Those parties make up about 20% of the parliament, whereas left-wing parties make up some 35% (with the rest being centrists and 'other').
If you think that Social democrats (S&D; center-left) are "left", then Christian democrats and conservatives (EPP; center-right) are "right". Those two are the traditional mainstream left-wing and right-wing groups in Europe. With these, we have 141 seats for the left and 178 seats for the right.
Then we have more radical parties with a clear position on the left-right axis. The inconveniently named The Left in the European Parliament have 37 seats, while their right-wing counterparts are ECR (66 seats) and ID (60 seats). This brings the total to 178 seats for the left and 304 seats for the right.
There are also two centrist-groups: Greens/EFA (72 seats) and ALDE (102 seats). The former is a weird amalgamation of greens, regional parties, independents, and pirates ranging from left to center. The latter consists of center to center-right parties that usually have some connection to the liberal tradition. But in some cases, the party in ALDE is more conservative and less liberal than their national counterpart in EPP. If we include these centrist groups in the calculations, the balance shifts further to the right.
Finally there are 49 MEPs outside the major parties, bringing the total to 705.
ALDE-PACE is basically Emmanuel Macron's party, so more socially liberal, and by that i mean he does accept that gay people do exist and can do whatever they want, if they want (the bar is low). They also are very pro-immigration in sectors that boost economies, but accept that immigrant workers can have equal rights. Extremely pro-Europe. I'd call them right-wing, but to be fair, only its leader is, most party members are pretty much center, center-right (they would be liberal-democrat in the US), and they push a lot of the legislation the greens want to pass, for multiple reasons (the green are seen as an "acceptable compromise", citing an EPP member i ate with).
I would not call the current Green left-wing either, its a torn party. I guess after the Covid and last summer, the wars and the resulting immigration, a lot of young people joined, and politically active young people are more left-wing, but the leaders are more center, center-left. But they hold major power on the left and can work with the other center party, and sometimes even the EPP. They are also on point (and have/propose good formations) with privacy and civil liberties, which might seems left-wing if you're in the US, but to me it's basically to political proposition of the old french party "les radicaux" which was so much in the center they split in two 30 years ago).
If you consider them like that, traditionally-conservative parties account for over 65% of current MEPs.
Oh, and that 'mostly'? Take it down a notch. There are no actual nazis in the European Parliament.
The "centrism" framing, btw, is fundamentally useless. In postwar Europe, PSE parties are left-wing and PPE parties are right-wing; other parties are fundamentally defined by their primary relationship with one of these two. The "centrism" mantra is reactionary twaddle to justify one's ideological vacuum.