Unfortunately, there is no way to get admitted to another university concurrently while receiving another degree. I recall when I applied to my PhD program that I was not allowed to take part in any other schooling or jobs for that matter. Also, funding should be a major concern. If you are on loans now, there is no way to defer those payments for 4 years, as most loan programs allow up to only 1 year of deferment for like an MBA or masters.
I do know a student that went outside the traditional MSTP program to find an advisor in a different field that was not traditionally offered as an MSTP type collaboration. If you enroll in an MSTP program you can likely complete a PhD in 4ish years, but a stand alone program can not guarantee that, as many standalone phd programs can vary from 5-7 years.
I hope this helps. I am very grateful I have my PhD, but I would seriously think twice about completing it again just due to the lost income of 4+ years of research. Also, when applying for positions, just having a PhD is a huge conversation point, and honestly most people say what field is your phd in, not where did you go. Even if your program has just a few advisors in a field of interest, then go talk to them. Your PhD topic will not define your career goals, but instead will give you a greater understanding and ability to do research in the future, regardless of the institution you attend.