Yes, Harvard Ed.D. = Ph.D.
Their Ed.D. is a very highly respected academic doctorate. And I know their Ed.D. acceptance rate is now around 7% (which is about the same as Harvard College admission rate); very difficult to get in. When they reject a doctoral application, they ask if you would be interested in a master's program instead. Their offer for master admission, even if you applied for the doctoral program, is then a polite circumlocution of rejection. (FYI, Harvard Ed.M.'s acceptance rate is around 50%).I did my Ed.M. at Harvard, so know a little bit about their doctoral program. I also heard that the research training required of their doctoral students is very rigorous and demanding. Overall, given that Harvard is usually ranked top in educational studies, a Harvard Ed.D. is one of the most respected research doctorates in the field (equivalent to, or much better than, education Ph.D.s from other universities).
Some Harvard Ed.D. holders are tenured professors even at non-education departments such as Stanford University's Business School or University of Chicago's Department of Sociology (U of Chicago is always ranked 1st in sociology). Some of them (whose concentrations are in counseling psych or developmental psych) got their jobs at psychiatry, pediatrics, and medical education departments at various medical schools including Harvard and Tufts meds. There is no doubt about the academic reputation of Harvard Ed.D. even outside of the field.