This is a broad(er) question, but I'll try my best to offer some biblical help. In Scripture, trust is not simply an emotion. It is faith and belief in the OBJECT of trust. When you are able to trust someone, it means you believe in their character. You cannot, for example, trust a thief or a liar. But you can trust a person who has proven to be honest, fair and faithful. Trust is only as good as the one you put it in. It must be EARNED.Of course, this applies to our relationship with God. When Isaiah says "I will trust the Lord and will not be afraid" (12:2), he means God is so utterly trustworthy that he does need to fear anything. God has Isaiah's good in mind, so whatever Isaiah goes through, he trusts in God.
This also applies to our relationship with our parents. How do you trust your parents? By believing in their character. Think of all the times in your past when your parents have come through for you. Think of all the sacrifices your parents haven made for you. When you are frustrated or angry at your parents, step back and know that your parents ultimately have your best interests in mind.
That being said, trusting a person is not easy. This includes your parents. Your parents are not perfect. They will make mistakes. They are sinners. That is why ultimately you must trust God first before trusting your parents. Know that God - unlike your earthly father - is perfect. He always has your goodness in mind. He allows things to happen to you to build you up. He will never abandon or leave you.
Consider what Jesus said in Matthew 7:9-11 (ESV):
Or which one of you, if his son asks him for bread, will give him a stone? Or if he asks for a fish, will give him a serpent? If you then, who are evil, know how to give good gifts to your children, how much more will your Father who is in heaven give good things to those who ask him!
Part of trusting in God means trusting that he gave you...your parents! Even with all their weaknesses and bad habits, they ARE your parents, and God instructs us to honour and obey them (Deut. 5:16). And honouring and obeying your parents means to trust them - even if you do not agree with them at times.
Hope this helps. In terms parents trusting their children, I believe the same principle applies. Children must earn the trust of their parents. They must prove their character is one of honesty, faithfulness and responsibility. Parents must also trust in God, and believe that God gave them their children as a gift. Ephesians 6:4 says, "Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord." This means fathers should not dominate or act harshly toward their children. Parents too must put trust in the children, bringing them up gently and correcting them lovingly when they are wrong. Children should have opportunities to prove themselves and to earn the trust of their parents.
[Answered by Pastor HM, who is learning to trust, but more importantly, learning to be trustworthy]
