You should note that the study you are citing is a 1993 study involving only 189 women over a three year period. See Tzonou A, Polychronopoulou A, Hsieh CC, et. al. Hair dyes, analgesics, tranquilizers and perineal talc application as risk factors for ovarian cancer. Int J Cancer. 1993 Sep 30;55(3):408-10. PubMed PMID: 8375924. Unfortunately, one limited study does not a theory make. To reach your conclusion, a much larger study is needed.
In 2005, a meta-analysis study published in JAMA reviewed medical studies from 1963 through 2004 to determine if there was a connection between any form of cancer and hair dyes. 210 studies were identified and, of those, 79 were relevant to the topic. The researchers concluded, based upon this evidence, that there was not a marked increase in the risk of cancer among personal hair dye users. This study identifies only two studies that show increase risk of ovarian cancer -- the study that you cite above and one involving occupational hairdressers. Obviously, the occupational study can be distinguished, and as noted above, the 1993 study is dated with a small population of women tested over a three year period. There simply isn't enough scientific evidence to claim hair dye causes an increased risk of ovarian cancer. To review the larger hair dye/cancer study see Takkouche B, Etminan M, Montes-Martínez A. Personal use of hair dyes and risk of cancer: a meta-analysis. JAMA. 2005 May 25;293(20):2516-25. PubMed PMID: 15914752.