Connection[]
Dib is the son of Professor Membrane. Whether or not Membrane's parentage is biological has never 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's only a temporary phase his son Dib will grow out of.
However, Membrane wasn't 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 on going to the "Family Dinner Night" without Dib, even though Gaz was ready to go, in "Bloaty's Pizza Hog". Also, 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 Membrane doesn't recognize Dib when he is 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."
Dib and Membrane in the Nightmare Realm
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. So, 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.
In Invader Zim: Enter the Florpus, Membrane expresses disappointment in Dib's continued interest in paranormal studies, and dismisses all his evidence as not real. Despite this, however, when Dib later states near the film's climax that he just wants to make his father proud, Membrane says that he's always proud of Dib. And then, despite thinking that the whole situation is a hallucination brought about by being hit on the head, Membrane doesn't hesitate to fight alongside Dib to protect him from Zim's Robot Army.
The second quarterly issue of the comic series has Membrane once again trying to be a good parent, as it's mentioned that he brought Dib along on his trip to Maxwell Beesworth's Cryptid Challenge so that the two could have a vacation together. When they're penned into their motel room by the media hounding Membrane over Beesworth's supposed Beeboogoat (actually a poorly-disguised GIR), the Professor expresses some disappointment over the fact that they won't be able to visit any tourist spots.
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 what he'd done wrong, but made Gaz happy.
Enter the Florpus shows how deeply frustrated and hurt Dib is by his father's refusal to believe him about his conflict with Zim, to the point that he's willing to hijack Membrane's Peace Day keynote speech just to prove himself to him by exposing Zim to the whole world. Later, after he's rescued Membrane from Moo-Ping 10, he's shown to be deeply touched by Membrane stating that he's always been proud of him, and being willing to fight alongside him.
In Quarterly 2, Dib is still driven to expose Zim to his father for the sake of earning his recognition. However, after events lead to GIR being mistaken for a Beeboogoat and Membrane becoming a laughingstock for refusing to believe it, Dib shows concern for Membrane's subsequent depression. Ultimately, when given a chance to finally expose Zim as an alien, he instead chooses to prove GIR wasn't really a Beeboogoat, restoring his father's reputation.