About halfway through the second season, their parents seem to have suspicions on them because suddenly Kyousuke has a close relationship with Kirino.
And at the end of the second season of Ore no Imouto ga Konna ni Kawaii Wake ga Nai, both of them are finally dating, even if just for a few months.
Did their parents know about this?
The later parts of Ore Imo contained all sorts of plot holes, dropped plots, and things that were just plain stupid. This was not the anime's fault—all of its plot holes and stupidity were copied straight from the light novel, but usually without the bloat that plagued the later novels.
Anyway, the novels give no clear indication about whether the Kousaka parents knew their children were "dating", but as @nhahtdh says, it seems very unlikely that Kyousuke would be allowed to live in the house if their father knew, since their father is also very protective of Kirino and often critical of Kyousuke. Manami did threaten to tell their parents, but relented.
As @noko mentions, there was a scene in both the novels and anime where their mother seemed to be on to something. (This was before they were "dating", though, so it actually was an unfounded accusation at that point.) This occurred in Chapter 1 of Volume 10 of the light novels. However, if we look at the conversation, it doesn't seem like she actually thought they were romantically involved:
Then Mom started with a topic that made me yell in surprise -----
"…Kyousuke, you…have you done something strange to Kirino?"
"Pfffff."
"Ha? Ha!?"
Hearing that totally unexpected topic, both Kirino and I were caught off guard.
"Mom? What did you just say?"
"I asked if you have done something strange to Kirino."
"What do you mean by 'strange'?"
"Yes! Strange means strange things!!"
Give me a good answer! You really are Kirino's mother!
"That…in other words…You are thinking that I have done some erotic things to Kirino?"
"Cough cough cough!"
Next to me, Kirino choked.
Well…her reaction is understandable.
Seeing my reaction, Mom's eyes suddenly became sharper.
"I didn't mean it like that, but since that’s the first thing that came to your mind, so it seems….."
All their mother does is ask if Kyousuke has done something strange to Kirino. Kyousuke himself brings up "erotic things", and their mother jumps on the fact that that was the first meaning he thought of for "something strange." She does go on to mention all the lovey-dovey things they've been doing recently, including coming home in wedding clothes, but until Kyousuke opened his big dumb mouth, she probably just wondered why they were suddenly so close and engaging in such strange behavior.
By the way, their father already knows about Kirino's otaku hobbies and how Kyousuke is in on it and covering for her. The novels remind us of that in this piece of dialogue:
Dad made a 'don't pass the ball to me now' expression then sighed and said:
"I'm not worried about the situation that your mother just talked about. I'm the one who understands what is going on between you and Kirino better than anyone. For you to cherish each other so much is within my expectations."
Dad, you are making it worse!
My face was gradually getting redder, my throat seemed like it was blocked.
"……"
No, no, don't say something so embarrassing! I'm embarrassed!
….I didn't even know why Dad spoke so highly of me ----- no, it’s probably because I was the one who brought Kirino back from America.
It's natural for a doting elder brother to take care of his little sister.
Now, I could say it without hesitation.
Still, Dad wasn't the type to just praise someone that easily.
"Kyousuke. Think back to about a year ago."
"A year ago?"
"Yes, about Kirino's - no, your hobby, we talked about it before."
"….."
Kirino's hobby – you mean the 'little sister based eroge'? When you caught her?
At that time, I insisted that it was my hobby.
This is probably enough to explain their newly found closeness for their father.
Towards the end of the novel, in Chapter 4 of Volume 10, Kyousuke asks his mother about her suspicions, and his mother replies that she never really believed they were romantically involved, reinforcing what I claimed above:
"Oh right, Mom. I got an A. So was your inexplicable misunderstanding solved?"
"What inexplicable misunderstanding are you talking about?"
"That — the misunderstanding that I had a love affair with Kirino."
Even now, I still don't understand. Mom suspected that I had an impure relationship with Kirino, so she forced me to keep a distance with Kirino. But why did she allow me to come back if I got an A? It had no effect on my relationship with Kirino at all.
"Ah, you meant that."
Mom burst out laughing.
"Actually, I never thought that you had done anything to your little sister."
"— What?"
Did I just hear something unbelievable? I thought about what I just heard —
"What! What is going on!?"
I roared and stood up.
"Recently, your relationship with Kirino has improved. So under these conditions, I separated you two. By doing that, you – the lazy brother would do your best due to the power of your siscon nature. Looks like my plan worked out in the end."
Mom continued praising herself, "I'm so smart."
The later novels don't ever suggest that their parents know anything, and it seems likely they don't. After all, as far as we can gather, not much changed in their relationship, at least externally. Maybe they're a little more lovey-dovey, but they were already pretty lovey-dovey by Volume 10. They were already going to each others' rooms at night and going out together on weekends; the only real difference is that the weekend outings are "dates" now. They do go around telling their friends that they're dating, but aside from Manami, none of their friends has any reason to rat them out, and Manami gave in, so it's probably just a sort of open secret among their circle of friends. Also, they only "dated" for around three months, and were probably rather scrupulous about keeping the secret during this period—no more coming home together in wedding clothes. So, all in all, it seems that the Kousaka parents didn't know anything.
