I am going to assume you are looking for a noun that refers to the situation you describe, and another noun for the action of claiming priority in that situation.
I am going to say that there is no such noun in general English. In parallel computing the analogous thing can happen with different processes, and then it is called a race condition. Occasionally that term will be used metaphorically for the kind of "who claims the ticket" scenario you describe in your comment, but most people wouldn't be familiar with the concept. And also it wouldn't feel natural at all to use it with either a love interest or a parking space (two concepts I don't usually lump together).
I think if the contest really just goes to whoever gets there first, you can say that it is a race for X. In the case of a love interest, I imagine getting there first isn't enough, and you could say a competition for X's affections (although that sounds a little old-fashioned).
I think the English term for calling dibs is calling dibs, or just calling something. You can also say that someone "lays a claim", "stakes a claim", or just "claims" something.