Prayer is something we tend to forget in a conflict. When we get into it with our parents, our general reaction is not to pray for help, but to fight more. James 4 says this:
"1 What causes fights and quarrels among you? Don’t they come from your desires that battle within you? 2 You desire but do not have, so you kill. You covet but you cannot get what you want, so you quarrel and fight. You do not have because you do not ask God. 3 When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures... 7 Submit yourselves, then, to God. Resist the devil, and he will flee from you. 8 Come near to God and he will come near to you. Wash your hands, you sinners, and purify your hearts, you double-minded. 9 Grieve, mourn and wail. Change your laughter to mourning and your joy to gloom. 10 Humble yourselves before the Lord, and he will lift you up."
We fight because we want something so badly that we are willing to hurt someone else for it. Most of the time we don't look at the conflict like this. We think we fight because the other person it being stupid or unreasonable. That might be true in some cases, but the way we try to resolve our differences is by trying to get the outcome we want. Instead, God wants us first to come to Him. We generally skip this step when we get into a fight, but this is what God asks us to do. He wants to give us what we ask for. He wants to answer our prayer. When we come, we don't ask that He just give us what we want, but that we could be used by Him, show His love, that what He wants will show through us. Our prayers are then answered and God does something amazing in the relationship. We now are not looking to fight, but to submit ourselves to God and have a conversation that honors Him. This is why we pray; it changes us, and God might just use it to start the process of change in the other person.