Why can't parents be godparents in Baptism?

In class on Friday, a very interesting question arose concerning the role of the godparent in a Christian Baptism.  I believe it was Annie who asked: "why can't a parent be their child's godparent or sponsor, in Baptism or Confirmation, respectively?" The reference to Confirmation of course only applies in those Christian traditions who observe it.  The class seemed generally puzzled by this question.  Speaking for myself, I knew that a parent could not be their child's godparent or sponsor, but was unsure of an answer other than that that was just the way it is.  
After class, I did researching on the topic, and it still proved hard to answer.  The most significant response to this question, I found on the Baptism FAQ page of St. Margaret Mary Catholic Community webpage of the parish housed in Louisville, KY.  The question reads: "Who can be a Godparent or a Witness?"  The site's full response is copied below (St Margaret Mary Catholic Community 2012). 

"You may have one godparent (sponsor) or two godparents (sponsors) for your child. Godparents will make a commitment to support the parents in their responsibility for the child’s religious upbringing. The godparents will serve as role models of the faith as they represent the larger church community. For this reason at least one godparent must be an active member of the Catholic faith. You may have one non-Catholic godparent but that person must be baptized in a Christian faith. It is important to note that while the godparents make a lifelong commitment to nurture the child’s relationship with the Lord, the Godparents are NOT legally responsible for the child if something happens to the parents. Separate legal arrangements must be made to establish guardians for the child in the event of the death of the parents. Insofar as possible, a person to be baptized is to be given a godparent who together with the parents presents an infant for baptism. A sponsor also helps the baptized person to lead a Christian life in keeping with baptism and to fulfill faithfully the obligations inherent in it. Can. 872 There is to be only one male sponsor or one female sponsor or one of each. Can. 873 To be permitted to take on the function of sponsor a person must: Can. 874 §1 1) be designated by the one to be baptized, by the parents or the person who takes their place, or in their absence by the pastor or minister and have the aptitude and intention of fulfilling this function; 2) have completed the sixteenth year of age, unless the diocesan bishop has established another age, or the pastor or minister has granted an exception for a just cause; 3) be a Catholic who has been confirmed and has already received the most holy sacrament of the Eucharist and who leads a life of faith in keeping with the function to be taken on; 4) not be bound by any canonical penalty legitimately imposed or declared; 5) not be the father or mother of the one to be baptized. §2. A baptized person who belongs to a non-Catholic ecclesial community is not to participate except together with a Catholic sponsor and then only as a witness of the baptism."

Under the Canon, the Catholic Church declares that one's parents cannot be their sponsor, shown be #5 above.  From this statement, I have concluded that one of the major reasons that parents cannot hold this position is to create a separation of responsibility for LEGAL and SPIRITUAL upbringing of the child.  The statement reads: "the Godparents are NOT legally responsible for the child," implying that the legality of the parents is separate from the religious upbringing of Godparents in this Tradition.  Perhaps also this separation provides the Baptisee to have another outlet for mentoring.  These appointed Godparents have a different relationship to the child than the parents.  Maybe this offset of responsibility was meant to ensure full attention in both legal and spiritual realms?  

Of course, this is all my personal opinion and not verified by any means.  However, I find it interesting that the answer to this question is so hard to find.  I also wonder if there is any variance in this 'rule' between Christian sects.  Any input?