When I am asked this question, I like to respond with the following analogy. Surgeons do not usually perform their own anesthesia, because anesthesia is a highly technical and specialized field, and the surgeon would prefer to leave that to an expert. By analogy, most researchers do not perform their own statistical analyses, because statistics is a highly technical and specialized field, and they would prefer to leave the statistics to an expert. So, if writing a dissertation is about learning how to do research, then by working with a statistician, you are gaining real-world experience in how to do research.
Another reason dissertation statistical consulting is ethical is, the statistical consultation can be an educational process for the doctoral student. That is, assuming you hire a qualified statistician that offers quality services (see my post on finding a qualified statistician). When I consult with a doctoral student, I offer unlimited email and phone support to ensure that my client fully understands their statistics. I consistently receive compliments for my ability to explain complex information in a way that is easy to understand. I have had several clients tell me that they learned more about statistics after talking to me for 15 minutes than they have learned in all of their statistics classes and text books put together. So, rather than the statistician doing all the work for the doctoral student, to the contrary, the dissertation statistics consult is a collaborative process in which the student learns about statistics.
Additionally, one other fact that shows statistical consulting is ethical is, many of my clients tell me their mentor suggested they hire a statistician. Also, I know of at least one university that posts links to dissertation APA editors and statisticians on their website. You can see that page here.