The short answer is no. There really isn't a paramedic job in the Coast Guard. Also, if you joined the Coast Guard, It would be hard for you to continue to get your CEU's and continue to work as a paramedic. Entering the Coast Guard, there will also be a lot of new information and qualifications that you will have to be spending your time on. It isn't possible to maintain your certification, it will just be on your dime and on your time.
There are some units that have EMT's and Paramedic's at that you may be able to take on some shifts but that would have to be a couple of years down the road in your Coast Guard career. As ratings go, AST's probably have the closest job to paramedics but we just have EMT certs. A couple of AST's are paramedics as well but they did it on their own time and they don't really practice that as being part of an AST. We are under the flight surgeon's medical license and he would have to authorize that. The Coast Guard probably isn't going to buy you paramedic gear and depending on what aircraft you are on, you wouldn't have space to bring it.
There are opportunities in the Coast Guard for you to develop medically but not any specific ratings. HS is mostly clinical, AST is EMT and although we train a lot for that kind of thing, the chance to use it just really depends.