Charles Helliwell's Answer:
OKÖwe've all had to deal with jealousy at work, and this one is no exception. However, it also sounds as if you have a double challenge in this instance in dealing with your Boss as well.
Your first task must be to extricate yourself from being on the defensive all the time. You are sounding like someone who is constantly having to justify your actions. Stop doing that. You have nothing to justify. You are in your position on merit and by right, so keep those thoughts at the front of your mind at all times.
Justification is always seen as the act of a defensive individual and consequently implies a negative and insecure person. There is a fine line between justification and validation. Validation of the decisions and choices you make, suggests confidence and self-assurance. Whether you feel it or not, you must exude confidence and self-assurance as a manager or supervisor. People are very quick to read and react to the signals of someone who isn't.
You are being undermined purposely by someone who wants you to fail. That is going to make you hyper-sensitive to any decisions you make or actions you take and this is going to make you start to question yourself in front of your colleagues. This is something you must not do. You must be decisive and self-assured at all times. Sometimes you'll make good choices; other times you'll make poor ones. Stand by those decisions and stand by your judgement, which will be sound.
Others less capable and less committed than you will always find fault with them. No matter whether you're the CEO or on the first step of supervision, that will always be so. Accept that and move on.
Now to your 'poisonous parasite'. Take her off-site and tell her directly that her behaviour is totally unacceptable and disappointing to you. Tell her that if she harbours ambitions to grow and develop and learn from you, then she is going about it the wrong way. Tell her that you expect her to be positive and supportive of you and the team, and that her actions are clearly not indicative of this. Tell her that you will be monitoring her behaviour and her commitment over the next six months and that you expect to see significant progress. Tell her that unless you see that progress, you will be taking actions to rectify this (you don't need to tell her what those will be. Let her stew over them); however, you would rather see her take a more positive and productive role.
Next, you have a conversation with your Boss and put him/her straight on the conversation you just had with the poisonous parasite. You tell your Boss that he/she put you in this supervisory position and that they are letting you down by not backing you more rigorously. Tell your Boss you are supervising the poisonous parasite and not them.
Consequently, you will expect your Boss to tell this person that they are to deal with you and not them. On the other hand, tell him/her that if they'd like to take on this person directly and manage them themselves, then they are more than welcome to her.
Your Boss sounds weak and lily-livered. (S)he won't want to take the risk of having this person report directly to them, because they will know how troublesome she is. And lastly, you must go and have a conversation with HR about the poisonous parasite and put it on the record, because at some point, you may have to start the process of exiting the individual.
In the meantime, stand tall; keep faith in yourself and your judgement and rally your department around you and your approach to the business and not her poisonous shenanigans.