According to this article from Creative Planet Network
"We have the technology, and our animators have the skills to do 3-D," says supervising producer Anne Garefino .
"We don't want it to look computery,"agrees director of animation Eric Stough , who's been on board since the pilot.
"We want it to look as crappy as possible."
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They do use 3D animation software.
However, from the beginning the cut-out paper look is what they wanted to achieve with CG after finding out the difficulties of using real paper cut-outs.
The decision to stay with the paper cut-out look was made from the beginning, they want everything to look as cartoonish as possible, down to the movements and appearance of the characters.
It also adds to the branding, one can immediately recognize a show of South Park by its looks.
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The show's style of animation is inspired by the paper cut-out cartoons made by Terry Gilliam for Monty Python's Flying Circus , of which Parker and Stone have been lifelong fans.
After the pilot for subsequent episodes have been produced by computer animation, providing a similar look to the originals while requiring a fraction of the time to produce.
Beginning with season five, the animators began using Maya instead of PowerAnimator.
The studio now runs a 120-processor render farm that can produce 30 or more shots an hour.
PowerAnimator and Maya are high-end programs mainly used for 3D computer graphics, while co-producer and former animation director, Eric Stough, notes that PowerAnimator was initially chosen because its features helped animators retain the show's "homemade" look.
On a different note, many times the creators do not want to opt for the latest technology for their movies or shows.
This could be based on personal like and dislike
and yes it is an artistic decision as mentioned correctly by Bobby Alexander in his comment.
A similar incident worth noting is Chris Nolan's choice of not shooting Dark Knight Rises in 3D at a time when almost everyone else was doing it.
Wikipedia says that Nolan elected not to film in 3-D, but instead stated that he intended to focus on improving image quality and scale using the IMAX format
So while South Park does use some of the latest technology, its current look and feel depends upon the creators' artistic decision.
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Would 3-D animation make the content any better?
George Lucas took advantage of the latest special effects available to film with the prequel trilogy which many fans say don't hold a candle to the original.
When he "remastered" the original trilogy with special editions, many fans revolted.
Look at video games, some of the most critically acclaimed games have simple animations and graphics (Minecraft, Fez, etc.), while there are countless graphic powerhouse games that are commercial and critical failures.
South Park is funny because of the content.
The barebones animation is a signature of the series.
Why change it?
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Also due to branding you don't want to change something that works.
The show is a screaming success.
Changing it might run the path when Coke introduced New Coke and failed miserably
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The quote I like best, referring to the software used to make South Park: "(using this 3D software) is basically building a sandcastle with a bulldozer".
That's the whole reason they use the software; to remain topical (episodes finished in a week, handed into Comedy Central on the Wednesday itself, ie tonight's episode wasn't submitted when you woke this morning!)
because they aren't bogged down with Michael Bay-ism.
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The technology has greatly improved since  South Park 's pilot episode, why has the style of the show not been changed?
Why are the Latest Episodes of South Park not in 3D animation?
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The decision to keep the paper cut-out look was a choice made from the beginning they want everything to look as cartoonsh as possibile and one can immediately recognize South Park by it's look. The decision to stay with the paper cut-out look was made from the beginning, they want everything to look as cartoonish as possible. 
The decision to stay with the paper cut-out look was made from the beginning, they want everything to look as cartoonish as possible. One can immediately recognize a show of South Park by its looks. They do use 3D animation software.