In a way there is, but it’s not exactly parallel. It’s the mitzvah to “Love your neighbor as yourself.” (Lev. 19:18)
Every person is a special and unique being who has three partners in his or her coming into this world. A person’s father and mother, who contribute the physical “genetic” components, partner with the Creator who adds the soul to the body.
Every person is important and a part of the design of Creation and deserving of honor and respect. However, the parents deserve an additional degree of respect from their children for at least a few reasons.
Gratitude. The parents brought the child into the world and nurtured the child. Out of gratitude the child should feel and display special honor for them.
Experience. Most people learn much from the myriad of events they experience in their lives. Parents in particular generally have specific learning experiences that they try to share with their children in order to raise offspring who will be independent and successful.
But I think the most important reason is that our parents – and each generation that precedes the following one – is an essential link in the chain of our Jewish tradition that goes back to receiving the Torah at Mt. Sinai some 3,300 years ago.
Parents are one step closer, one link closer, to that pivotal event that turned us into the Jewish nation. Each of these links is of paramount importance as it serves as the means to continue the tradition from one generation to the next. In addition, parents are one step closer than their children to the original handiwork of the Creator when God made the first people, Adam and Eve.
A true story to illustrate the importance of our parents serving as links in our tradition: In response to an anti-Semitic comment made by Daniel O'Connell in the British parliament, Benjamin Disraeli memorably defended his Jewishness with the statement, "Yes, I am a Jew, and when the ancestors of the Right Honorable Gentleman were brutal savages in an unknown island, mine were priests in the Temple of Solomon".