If the field is conservative, then parametrization is not necessary; the work will actually be independent of the path, and you will be able to find a potential that you can use to find the work.
If the field is not conservative, then the work will in general depend on the path you use, and you must somehow parametrize whichever path you use. Even if the field is conservative, you might still prefer to pick a path and parametrize it (that might be easier than finding the potential, for example).
I am not going to say if this particular example is conservative. This appears to be a pedagogical problem. The point of the problem is to get the work along a few different paths, and compare them to see if they are the same. For the purposes of this problem, parametrization each path is the only way to solve it.