I have thoughts but can't create words, but going to try.
Why both? I can see where you might want to get into the combined public health aspect where both humans/animals are affected. Defintiely an MPH and even further school in public health would be the route to go. Honestly, not sure a DVM or PA would be needed to jump into a public health career if that is really an interest. Would save you a lot of schooling, money and years to have a direct focus.
I won't say it is impossible to come out and work 8 years in vet med and then go to PA school, of course, you certainly can. Anyone can change their mind at any time and choose a different career path. I just don't think it is going to be feasible for you to do both.
You generally aren't going to qualify for benefits if you are working two part time jobs. So no health insurance, minimal (if any) paid time off, vacation leave, sick leave will be less. No retirement plans/benefits. Short-term/long-term disability are much different for part time vs full time employees. PLIT/licenese defense fees may or may not be covered.
Do you have any idea on how relief works? It varies depending on what you are looking to do. There are some clinics that will add in the relief vets as w-2 based employees, but these are not near as common. Basically being a relief vet is like creating your own independent business and you are the business owner. You often will want to develop an LLC, hire a tax person and need to be prepared for the income tax differences (you have to pay both the employer and employee portions of income tax, in general). It likely wouldn't be of much benefit if you are planning on doing very intermittent DVM shifts while attending PA school.
Plus these are two completely different careers. I mean, you don't just go to vet school and bam, done, no further learning needed. There are consistent new medications, products, scientific developments, etc. Are you going to keep up CE and license fees while attending PA schoool? I mean you will have to if doing relief work while attending PA school. Are you prepared to pay for all of the career fees of a DVM while doing minimal if any veterianry work? Don't forget professional liability insurnace and license defense and these will not be covered as a relief vet by any clinic you relief at. If you are not prepeared to financially and time commitment wise keep up on licenses and CE, are you aware of what would be needed to reinstate a veterinary state license after a period of not practicing and not keeping up on CE? I am sure this varies by state. Do you have a back-up for paying for PA school that is not veterinary relief work?
Then you have any professional fees/dues associated with being a PA. I am not sure if they also have CE requirements or not, I would think they do. So you will now have two careers in which you have to upkeep, licenses, different types of CE, professional fees/dues (AVMA, PA groups, state vet groups/PA groups, etc, etc).
Not to mention that the first few years of any medical career is a giant learning curve and significant career growth and development. At 8 years of vet med, you are likely going to be comfortable with most things, probably be fairly grounded at this point, you will still have some things that you need to look up/investigate further, but you are going to be at a point in your career where things are really finally starting to settle in and feel more comfortable and then you are going to ditch it all to go do an entirely different career (of which you might need to repeat undergrad for) and effectively start over. Excpet if you are doing PA only part-time initially you are going to have a much longer time-frame to really settle into the career. Not to mention when you get back to vet med, you will have another learning curve again (albeit it will hopefully be less intense), but I can promise there will be new medicaitons/treatments by the time you'd get back.
I simply jumped to a different area of vet med a couple of years ago and holy carp did the comfort level go from fairly comfortable with the daily routine to "the heck am I doing?" rather quickly. I can tell now the learning curve is plateauing out again a bit, but I am still intensely learning new things daily. And that was just jumping to a different area of the same career field not attempting to do two entirely different careers at the same time.
I will be honest, I do not see how anyone can be a DVM and a PA on a part-time basis and actually do both at peak performance/do them both well. One or the other, or both will suffer in some way.