Rank advancement, including merit badge requirements, are signed off by the Scoutmaster, or someone that the Scoutmaster designates. In Troop 990, this is a registered adult leader, and almost never the parent of the Scout. Parents are excluded from signing their own son's advancement to avoid the appearance of favoritism and to strengthen the mentor relationship between a Scout and adult leaders.
So, what is the parent's role in their son's advancement? Parents are encouraged to teach their son(s) Scouting skills, both individually and in groups. It is a great help for a Scout when his parent can help him learn his knots, or to help set up a tent in the back yard two or three times until the Scout becomes familiar with the assembly. It is outstanding when a parent can attend a campout and assist in a five-mile hike or a fifty-miler backpacking trek. This participation helps not only the parent's son, but other Scouts that are involved in the activity. This is one of the many opportunities a Scout has to see cheerful service being offered by adults.
The mentor relationship between a Scout and adult leader is important to a Scout's development. This relationship will be similar to that between him and his supervisor at this first job, and is like the better teacher/student relationships at school. An important difference is that these people will help the youth with facts and skills, while the Scouter helps the Scout learn to make ethical and moral decisions in his life. Parents plays critical roles in their son's development, but it is difficult for a parent to successfully fill the parent and mentor role.
You can see the word almost used in the first paragraph. There are a few instances where a parent is one of the two-deep leadership, instructing a group of Scouts. It is always preferred that if two-deep leadership includes one adult that is not the Scout's parent, then that leader should sign for the Scout. With the Scoutmaster's approval, a parent can rarely approve his or her son's advancement. This should always have the Scoutmaster's approval before the activity begins.