Let me start by giving you full disclosure in that I am a professional counselor/therapist.
One of the key questions in dealing with any anxiety or behavioral situation is to assess how the "problem" affects or hinders your normal functioning. In other words, is this really a problem or is it just a strange characteristic about me that does not overall negatively affect me. Can you give me some more information about your specific situation?
For example, do you obsess about this regularly and does it make it difficult for you to get through your daily routine because it's always "on your mind" or is this just something you wish was different, but you don't ever really think about it until you find yourself in public and you "need to go"? What is the experience of being in public and "needing to go" like for you? Do you just put it off and tell yourself "I'll go later" or are there physiological responses like breaking out in a sweat, becoming jumpy/irritated, or losing the ability to concentrate/focus. Determining how this affects your functioning is a way of assessing the severity of the "problem" and will help indicate what level of help, professional or not, that you need.
And, as was stated earlier, in every situation you must rule out that this is not a physical or medical condition. Are you able to use the bathroom normally at home without any difficulty?
Sorry to keep going, but you also mentioned that you were the victim of teasing/bullying at a younger age and that these experiences often occurred in the restroom. It is possible that you are experiencing a cognitive distortion in that you have paired the emotions associated with past trauma to a routine life activity. If this is the case or if the bullying was in actuality abuse, then professional help would be extremely beneficial in helping you overcome this.
No matter what, suck it up and tell your parents.

I don't know your family relationships, but I would hope that if you approach them from the perspective of "I need your help with something" that they will lovingly respond in any way they can to support you.
Let me know if there is anything else I can do to help or if you have any other questions. Be positive, you
can do this.