All left-right political terminology is inherently subjective, so you can argue neoliberalism is promoted by center-left parties as long as you’re defining the center as being to the right of that. Since this post seems to be about the United States, that center is already pretty far to the right as measured from, say, Denmark (picked a name out of a hat). I think the bigger argument here is about US-defaultism rather than whether or not it’s OK for Americans to describe things in terms that relate to their political climate.
EDIT: I think the comment I’m replying to is confusing people. Replying solely to the words “center-left” makes it seem like the OP described neoliberalism as center-left, which people are objecting to. However, the OP only used the phrase center-left once, to say that American center-right and center-left parties have enacted neoliberal policy. As a statement of fact, the Democrats have enacted neoliberal policy. By American standards, the Democrats are regarded as center-left. This does not mean the OP was saying “neoliberalism is a center-left ideology.” There is an argument to be made here that the Democrats are not a center-left party, but I think the issue is getting confused here because people are reacting as if the thing being described as “center-left” is neoliberalism, when it’s actually the Democratic Party.
Of course it’s subjective. The terminology of the left-right political divide originally referred to 18th-century France. In the 21st century, we’re usually not defining the political center of a nation by how it compares to the French Parliament of 250 years ago. The center moves over time and space, and the left and right are relative to that center.
I do think this comment thread is confusing people, though, as noted in an above edit. For clarity, nobody is saying neoliberalism is a center-left movement.