If one's parent is ill, one would probably want to visit. It is, it seems, some level of mitzvah (though it isn't explicitly on the Rambam's list of 613, it is included under some other general category).
However, honoring parents is also a mitzvah (number 59 on the aforementioned list). The Aish website includes this statement:
In general, a child should be eager to fulfill his parents' wishes. There are some limits, however:
This first limit is
If a parent instructs a child to do something that violates Jewish law, the child should respectfully refuse to do so 
 Yoreh De'ah 240:15, based on Leviticus 19:3 with Rashi. This includes posthumous requests, such as to cremate the parent's body (Achiezer 3:72, based on Maimonides – Avel 12:1), or not to say Kaddish (Yabia Omer, vol. 6 – YD 31:4).
This concept of the conflicting obligations, kibbud vs. keeping the mitzvot is discussed in general here.
What happens in a case where a parent expressly states that she or he does not want any visitors when sick -- no calls or anything else -- the parent wants to be alone with his/her thoughts because that will effect the most efficient recovery.
Is "not visiting the sick" a transgression (or simply the missing out on an opportunity to fulfill a mitzvah) and enough of a transgression that it cannot be demanded by a parent?
Does the existence of it on some rabbinic level make it "real" enough that a child must perform it even in the face of a parental wish otherwise which is bound up in a clear de'oraita mitzvah?