I'm not sure that it's in fact harder to become a derm as a DO vs. as an allopathic medical grad. It would be MUCH harder to get into an allopathic derm program, and perhaps impossible. However, DO's have their own derm programs. I don't know what the match rate for a DO student into a DO derm residency would be...it might be less, the same or better than the match rate of an MD grad into a MD derm residency.
If the OP is asking whether it will be harder to get a given allopathic residency spot than if he/she attended an MD school, then I'd say the answer is yes. However, DO's have their own own residency programs, including in specialties of surgery, etc. I am not personally familiar with these, or how hard it is for a given DO student to get a spot in a DO program in something like radiology or urology or derm (specialties that are very competitive to get in the MD residency match). A DO trying to get an allopathic residency in something like internal medicine, family practice or pediatrics will not in general have a hard time, but some allopathic programs will not take you. On the flip side, NO osteopathic residencies will take any MD students, so you have those all to yourself.
If the OP is asking whether patients and other doctors are going to discriminate against him/her in practice by virtue of being a DO, I'd say the answer is no the great majority of the time. Only docs who are jerkoffs would be likely to care what your degree is, especially in the future (as it tends to be more the older MD docs who care/discriminate either). They will care whether you act nice, work hard, and appear to be competent. Ditto for patients, plus most patients are clueless about medical training and don't know what a "medical doctor" is either. One I talked to recently thought that "medical doctor" means that the doc is a primary care doctor (rather than a surgeon, radiologist, etc.).
If you were able to get several MD interviews, that suggests you'd have a shot as a reapplicant to MD programs, but you have to think about how strong a shot you'd have. Also, if you are a California resident, those state schools are notoriously hard to get into, so you might be taking a big risk by giving up a guaranteed admission at the DO school. I think it's important whether you liked the DO school when you interviewed, and also if you can find out if the graduates from there are able to go on and match well, and do the things you think you might be interested in, career-wise. I wouldn't give up the DO admission just because you fear some nebulous snobbishness by colleagues at a future date, b/c I don't think it's going to affect you much.