Trust is a two way street; you get it if you give it. For example: if you want your parents to trust you try trusting them with the truth about what is going on in your life, if you want a relationship built on trust you have to trust in the other person as much as you do in yourself.
Parents will usually give trust freely until you do something to break it. If this isn't the case in your family and you really can't see why your parents don't trust you just come right out and ask them for an explanation.
Sometimes people don't trust you because of the actions of a person close to you. It is all too common for the behaviour of older siblings or friends to cause parents and teachers to doubt you. Accept that this is what is happening, don't fight it, and work at proving that in spite of the actions of others you can be trusted.
Sometimes trust has to be earned. If you had trust but did something to break it, it is possible to fix things. But in doing so remember that as the trust breaker you don't get to set the expectations, terms or time frame for earning that trust back.
In order to fix broken trust both sides have to want the trust back. You can never force a person to trust you.
Your parents want to trust you! What you see as mistrust of you could really be fear, mistrust of those around you and/or their natural protective instincts at work.
Some people have been so badly hurt in the past that they have great difficulty trusting others. If you have a person like this in your life accept that there will always be an element of mistrust in your relationship and be prepared to work overtime to prove that you can be trusted.
Trust is an essential part of ALL successful relationships be they academic, romantic, friendly or familial.
Trust is an essential part of sex - if you don't trust your partner 110% don't have sex with them.
Trust is a gift - you give it and you receive it. It should never be taken for granted.