The Prophet (sallallahu alayhi wa sallam) said:
"Make the orphan come close to you, and be nice to him, and wipe his head, and feed him from your food. That will cause your heart to be soft, and your needs to be fulfilled." (Tabaraanee)
Once Anas, ibn Malik (RA) was asked by someone to make duaa for him. He replied,
"Verily, duaa is raised up (to Allaah) by good deeds," (al-Awaayishah, p55)
indicating that the questioner should strive to do good deeds if he wants his duaa to be answered. And Wahb ibn Munabbih said:
"The example of the one who makes duaa without doing any deeds is like the one who tries to shoot arrows without a bow." (Ibn al-Mubaarak, a1-Zuhd)
So the person who wishes to have his duaa responded to should increase the quantity and quality of the good deeds that he does. One of the best deeds that a person can do, in fact, the most important and the greatest act of worship after worshipping Allaah, is that he be dutiful to his parents.
The rights of parents are indeed very great, and it is because of this that the person who fulfils these rights earns a very high status with Allaah. Of the blessings that such a person has is that his duaa are answered.
This is indicated in the Uways al-Qarni's hadeeth, who among the tabi'ee has the highest status of all. Umar ibn al-Khattaab narrated, I heard the Prophet (salallahu alayhi wa sallam) say:
"There will come to you (a man by the name of) Uways ibn Aamir, with some people from Yemen. He is from (the tribe of) Muraad, then from Qarn. He was afflicted with leprosy (all over his body) except for one part, the size of a dirham. He has a mother to whom he is very dutiful. If he were to swear by Allaah, Allaah would fulfill his oath. If you are able to ask him to seek your forgiveness, then do so." (Muslim)
When Umar heard this, he waited until Uways came to Madinah, and then asked him to make duaa for him. This hadeeth shows that being dutiful to one's parent is a cause of one's duaa being accepted, as the reason that Uway's oath was responded to was because of the devotion that he used to display to his mother.
Also, the story of the three people that were trapped in the cave further proves this point, since one of them mentioned his good character to his parents as a means of tawassul (approach to Allaah).