Honoring parents is a form of that honor we owe all authority, including God himself. The Israelites were commanded to revere God (Deut 10:20); that same word was used with respect to parents (Lev 19:3). He is our heavenly Father, and we are his children. "A son naturally honors his father and a slave respects his master. If I am your father, where is my honor? If I am your master, where is my respect?" (Mal 1:6). As John Calvin said, "The dishonoring of parents redounds to the dishonor of God himself, nor can anyone despise his father without being guilty of an offense against God."
Obey your parents in the Lord, for this is right (Eph 6:1; cf. Pro 6:20-24). This is perhaps one of the most fundamental ways in which we can honor our parents. Although now, as an adult, your parents are probably not telling you what to do anymore, there are still ways in which we can respect the spirit of this command. For instance, they might have certain expectations of behavior in their house, so we comply obediently by ensuring we and our family adhere to those expectations, even if we don't agree or practice them in our own home. That is love in action.
Speak well of your parents, or at least refuse to speak evil of them with others, including your immediate family, church family, co-workers, and friends (and maybe even to yourself). Don't seek solace or validation in being a victim of their misdeeds or mistakes and airing that dirty laundry. Under the old covenant, to castigate, smear, or speak evil of or abusively toward one's parents was punishable by death (Exo 21:17; Mark 7:10). We should speak well of our parents, especially in ways that edify and support them.
Seek the advice and opinions of your parents, which communicates that you value them. With their experiences and the mistakes they've made along the way, they probably have some wisdom to share. Even if you have parents who give questionable advice or opinions, ask them anyway just for the blessing that it will be to them. Asking does not obligate you to actually take their advice or agree with their opinion, but asking communicates love and respect.
Don't nurture suspicion or resentment toward your parents, holding their misdeeds and mistakes over their head. Remember, love is not malicious or disposed to find fault, nor does it treasure resentment or impute improper motives to others (1 Cor 13:5). Honoring your parents involves putting the best possible construction on their opinions, motives, and conduct, choosing to think well of them and being forgiving of their faults.
Remember your parents as you grow older and perhaps form a family of your own. As the demands and stresses of life amass over the years, we can tend to forget our parents more often. Do not forsake your parents in their old age. Return the love, nurture, and support they gave you throughout childhood by calling them, writing to them, remembering their birthdays and anniversary, visiting them, having family get-togethers and so on. A Christian's parents should not suffer emotional neglect and abandonment.
Support your parents and provide for their needs, for those who are able but refuse to provide for their own family, including their aged parents, have denied the faith and are worse than unbelievers (1 Tim 5:4, 8). As William Barcley said, "The raising of children requires tremendous sacrifice, and it is only right that children make sacrifices for parents in return."
No, admittedly, I was writing from a Christian perspective. But what I was saying has roots deep in Judaism (Wikipedia link)—which makes sense, given the prominent place the Tanakh (Old Testament) and ancient Hebrew faith has in our canonical scriptures and theology. That link I provided there has a fair amount of Jewish support for the points I was making (e.g., Talmud).