I would check your SD's policies. In ours, for example, parents or teachers can start the process. However, it is very regimented and full of deadlines. Usually, you have to apply in late November, if kid seems qualified, they go to a group setting achievement test. If they test at a certain level, then they move on to the CogAT, and so on.

I have heard of very few parochial schools that have gifted policies. I would talk directly to the teacher and/or administration.