Whom do I serve?
How do I honor my parents when their will is in conflict with God's will?
Good question, Anna, but hard to answer directly because of all the variables involved.
Your age, your dependence upon your parents, the level of mutual confidence, antecedents (former experiences), and the situation you are facing.
1) Listen carefully to what they say. They know you and undoubtedly love you and want the best for you. Try to understand where they are coming from.
2) Talk to your pastor or a mature adult Christian, and explain the situation and ask their advice.
3) Talk to the Lord about the situation. Read the Scriptures daily and ask the Lord to give you wisdom and guidance.
4) If your parents want you to do something that is clearly against your understanding of Scriptural morals, you can't obey them. You have to "serve God rather than man." However, if it is a matter of timing (you feel it is God's will for you to participate in some event or make a life choice of a career, etc.), you may find the Lord telling you to be submissive to them, at least for now.
(Note Jesus' example, before he began His public ministry: "He went down to Nazareth with [his parents] and was obedient to them" (Luke 2:49).
5) However, the time may come after an appropriate interval of time when you feel you need to go off on your own if your parents won't allow you to do what you feel the Lord is calling you to. Jesus did say that we are to love Him more than family. But I wouldn't take this step without the counsel and support of several mature Christian friends.
Your situation is a painful one. Put all the details before the Lord, and as you wait patiently He will guide you. It's His promise.
"If you want to know what God wants you to do, ask Him, and He will gladly tell you, for He is always ready to give a bountiful supply of wisdom to all who ask Him" (James 1:5)