OK...We 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(release) 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 your 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 undermined purposely by someone who wants you to fail.This is going to make you hyper-sensitive to any decisions you make and actions you take and this is going to make you 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 will make good choices; and others times you will make poor ones.Stand by those decisions and stand by your judgement, which will be sound.
Other less capable and less committed than you will always find fault with them. No matter whether you are 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 that her behavior is totally unacceptable and disappointing to you.Tell her if she harbors 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 an the team, and that her actions are clearly not indicative of this. Tell her that you will be monitoring her behavior in next six months and 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 you boss that she/he put you in this supervisory position and that they are letting you down by not backing you up more vigorously.Tell your boss that your are supervising the poisonous parasite and not them. Consequently, you will expect your Boss to tell this person that she is to deal with you and not them. On the other hand, tell him/her that if they'd like to manage this person themselves, then they are more than welcome to her.
You boss sounds weak and lily-livered(cowardly, timid).(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 and put a record on it, because at some point, you may have to start the process of exiting this individual.


In the meantime, stand tall; keep faith in yourself and you judgement and rally your department around you and your approach to the business and not her poisonous shenanigans(trickery or deception).