As part of the Ten Commandments, God commanded that children honor their parents:

Exodus 20:12

12 Honour thy father and thy mother: that thy days may be long upon the land which the LORD thy God giveth thee.

This commandment also appears in Deuteronomy 5:16, Matthew 15:4-6, 19:19, Mark 7:10, 10:19, Luke 18:20, and Ephesians 6:2-3.

However, there are statements that the Lord made while He was here on earth, which are construed by critics of the Bible as contradicting the command to honor one's parents:

Matthew 10:35-37

35 For I am come to set a man at variance against his father, and the daughter against her mother, and the daughter in law against her mother in law.
36 And a man's foes shall be they of his own household.
37 He that loveth father or mother more than me is not worthy of me: and he that loveth son or daughter more than me is not worthy of me.

Luke 12:51-53

51 Suppose ye that I am come to give peace on earth? I tell you, Nay; but rather division:
52 For from henceforth there shall be five in one house divided, three against two, and two against three.
53 The father shall be divided against the son, and the son against the father; the mother against the daughter, and the daughter against the mother; the mother in law against her daughter in law, and the daughter in law against her mother in law.

Luke 14:26

26 If any man come to me, and hate not his father, and mother, and wife, and children, and brethren, and sisters, yea, and his own life also, he cannot be my disciple.

All of these verses teach the same truth, and so they can be dealt with together. They all teach that if you ever have to choose between Jesus Christ, and your family - including your parents - you are to choose Jesus Christ.

Depending on a person's parents, this may be more or less frequent. If your parents are godly, they will not be frequently putting you in a position where you must choose them or Christ, because their wills will be in agreement. However, if that choice needs to be made, Jesus Christ is commanding you to disobey your parents, if they command you to do something sinful, or disregard your family entirely, if they are standing between you and living the way that Jesus commands us to live.

This is not a new commandment, nor does it only apply to the God-sanctioned authority structure of children obeying their parents. It also applies to the God-sanctioned authority of civil government (Romans 13:1-7), and wives being in subjection to their husbands (Ephesians 5:22-24, etc.). Under normal circumstances, if those authorities do not contradict God, you are to honor them, because they are legitimate.

However, over all of those authorities is God, and so fulfilling the commandment of "Let every soul be subject unto the higher powers" (Romans 13:1) demands that a person always chooses to obey God whenever there is a conflicting authority telling them not to.

Naturally, this is going to cause division in families, as Christ mentions in the passages above (Matthew 10:35-37, Luke 12:51-53). However, God is not the root of the division. The division is caused by wicked, ungodly family members, who are not going to tolerate one of their family member's Christianity, because they themselves reject the Lord. If everyone fulfilled what the Bible demanded of them, every family would be a peaceful, loving, edifying unit. It's sin that causes division.

All of this explains Jesus's command to "hate" father, mother, yourself and so on, in Luke 14:26. This is, obviously, not a random, indiscriminate hatred. This is instead a rejection of any pressure to compromise the truth, or disobey God, whether that comes from your parents, your spouse, or the ambitions and desires of your own flesh. It is a command to despise any will that conflicts with Jesus Christ's.

Therefore, as a general rule, children are to honor and obey their parents, wives are supposed to honor and obey their husbands, and people are supposed to honor and obey the civil authorities. However, all of those authorities must be utterly rejected in favor of Christ, if they ever come into conflict with what Christ commanded.