Connection
Dib is the son of Professor Membrane. Whether or not Membrane's parentage is biological has not been confirmed; however, it has been said that later in the series, it would have been revealed that Dib was created by Membrane in his lab, possibly as a clone or near-clone of himself. Nevertheless, the fact that he shares similar traits to his father indicates that Dib (and perhaps Gaz as well) were created from their father's own genetic material. Thus, they are in a sense his biological children.
Membrane
Father and son.
The Professor is a man of science and logic, so he finds Dib's obsession with the paranormal silly, and believes it is only a temporary phase that Dib will grow out of. However, Membrane was not surprised when Dib was committed to the Crazy House for Boys in the Halloween episode. He even went so far as to say "So, I find you here, son. I suppose it was only a matter of time. You know better than to use my tools for your parascience!"
Membrane is very embarrassed by his son's interests, especially when foreign delegates came to visit his house in "The Sad, Sad Tale of Chickenfoot". However, these moments of shame are usually very brief, because most of the time Membrane is so busy he forgets he even has a son. It could be argued that Dib's sister, Gaz, is the "favorite child". However, it could also be implied that he cares for both of his children the same, since he refused to go to the "Family Dinner Night" without Dib, even though Gaz was ready to go, in "Bloaty's Pizza Hog" and he overcompensated repairing the harm made to Dib with an exosuit and weapons in "Bad, Bad Rubber Piggy".
There are several times in the series that he doesn't recognize Dib when he is
Dib and Membrane in the Nightmare Realm
standing right in front of him, instead calling him "little boy." Most of the time, Membrane isn't around, and merely has a computer watch his children. There is only one day a year that Membrane spends time with his son, their annual "Family Dinner Night."
In the first issue of the comic series, Membrane appears to be acting as a much more active parental figure, such as insisting Gaz give the shut-in Dib his dinner, and grounding him for staying out all night. However, later issues swing back and forth on this - in Issue 5, Membrane knowingly aids Gaz is finding an alternate universe ruled by gamers she can enslave and torture Dib in, just because she says it's "for science", while his alternate universe counterpart chooses Foodio 3000 over Dib when the gamers comes for him. In Issue 12, a message from Membrane's future self reveals that he left Dib behind when he and Gaz fled Emperor Zim's conquest of Earth (albeit on Dib's insistence). In Issue 14, Membrane reveals to Gaz that Dib has accidentally swallowed nanoexploder robots and will explode if he gets upset, and therefore Gaz must be nice to him and keep him happy; later, when the family are at dinner, Membrane says he's happy to be there. In Issue 16, Membrane gives Dib an experimental Fear-Siphon to make himself fearless at his request, even though he knows it's not yet ready for human use.
Dib
Dib doesn't seem to mind that his dad sometimes ignores him, and often counts on the fact that Membrane thinks he's crazy, as said in "Gaz, Taster of Pork". However, there are still times when Dib wishes his father would remember him, like in "Battle-Dib" when he needed a signed permission slip.
Dib has a very strong sense of priority, so he finds the annual "Family Dinner Night" with his dad much less important than saving the world, and even plans on going on a mission into Zim's base the night of the event. In the unfinished episode "Mopiness of Doom", Dib quit paranormal investigating to work with "real science", to the Professor's delight. Dib was overjoyed to have his father's praise and attention, but eventually gave it up in favor of paranormal research and stopping Zim which greatly upset the Professor to the point that he would have cried and ask himself why he did wrong, but made Gaz happy.