Absolutely yes, women are allowed to drive in Bulgaria, the simple answer is, there is freedom to do some things, it is a part of their right to have possessions and to use whatsoever they have. Just like in other countries of the world, I don't think there is a law stopping or restricting women from driving unless the decision not to drive is borne out of one's will.
There are many husbands who are over protective, they might urge their wives not to be driving cars because of her safety. But if you are driving car regardless of the gender, you must obey the driving rules, you don't drink while driving and you must also obey traffic rules. Kids that are below 12 years of age are not allowed to be staying in the front seat, they stay at the back seat.
All these rules must be kept when driving on the road, lest you are going to face the consequences for disobeying the law.