Was there a production rule forbidding the depiction of parents in Disney shows such as Ducktales?

Note: I am ignoring movies where the main plot revolves around the main character reconnecting with their parent; because these movies are inherently focused on the relationship between the child and parent. This is about movies where the plot occurs independently (the vast majority of movies), where the parental relationship is optional plotwise but seemingly often omitted from the movie; which is what your question is about.

Most Disney main characters are children (or young adults who are still childishly naive). This is because children are the intended audience; and the viewers tend to relate to the main character. The movie's plot teaches lessons to the main character, and by extension the child that watches the movie.
Parents are often a mentoring presence to children. They are safe havens. If the child cannot talk to his parents, it must always be because the child cannot possibly talk to their parents.

Parents are the antithesis of plot development; as parents provide solutions and plots are there to provide problems. The main character needs to be struggling, because the movie plot requires them to.

There can only be three reasons why a child cannot talk to its parents, and two of these should generally not be shown in a lighthearted movie for children.

• The parent refuses to talk to the child, even though the child wants to. You cannot show this in a kids movie because it is a pretty dark subject matter to a 5-7 year old. Comparatively, a child (as the viewer) can comprehend not having a parent better than having a parent that refuses to speak to you.
• The child refuses to talk to the parent(s), even though the parent wants to. Although nothing stops you from showing this in your movie, I highly doubt that the parents in the audience will be willing to take their children to movies that teach them about disobeying your parents.
• The child and parent cannot talk, even though they both want to. This can be present in many ways: a dead parent (e.g. Yasmin and Aladdin), a child that is separated from its parents for the duration of the plot (e.g. Alice in Wonderland), a parent that is unavailable because of external reasons (e.g. Aladdin's dad in the third Aladdin movie). The main key is that this is beyond the child's control; and therefore the child is alone (at least during the movie if not permanent).

Since a healthy parental relationship is counterproductive to plot development, you can't really display a dysfunctional parental relationship, the only option that remains is where the child/parent are a slave to their circumstances. They want to talk to eachother, but something prevents them.

Now we get to the third and last component of this: Checkhov's Gun. Briefly explained:

Chekhov's gun is a dramatic principle which states that every element in a story must be necessary, and irrelevant elements should be removed.

In the Lion King 1.5: Hakuna Matata, Timon only has a mother. He was originally slated to also have a father, but the father character was scrapped and his lines were given to Timon's mom.
The reason for this was relatively simple: it reduces the amount of characters there are in a movie. Especially for children, too many characters in a movie would make everything too confusing.

Timon's mom doesn't even appear all that much in the movie. She's in the first 5-10 minutes, and shows up again in the end. But she isn't really involved in the plot, other than justifying why Timon is the person who he is.
Having to remember both Timon's mom and dad would be an unnecessary complication that doesn't particularly add to the movie or its plot.

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So to combine these answers:

Parents are often counterproductive to plot developments, and as a result will often be unavailable (and irrelevant) during the events that take place in the movie.
Their presence in the movie is therefore not needed (since they do not influence or contribute to the plot), and they are often omitted in order to keep the movie focused on only its plot, instead of sidetracking the movie with unnecessary details.