The Cosby Show ran from approximately 1984 - 1992, won 4 Golden Globe Awards and 48 wins and 47 nominations (imdb.com).
Seen below are the four rugs we take a look at.
Above, starting up top and moving counter clockwise: Green arrow points to what appears to be an isfahan (also spelled Isphahan) design rug. Orange arrow points to what appears to be a Kerman design rug 4'x6' in size. Blue arrow points to what looks like an antique tabriz design. Finally, the red arrow points to what we believe is a Kashan design area rug.
While the images are not particularly great, we can see the area rug directly in front of the doors (seen above) to their home has a medallion similar to those found in Kashan oriental rugs. Without actually inspecting these rugs, we can say all the rugs seen are all what one would consider a "Persian Design," however identifying whether or not these are actual rugs from Iran is practically impossible without high resolution photographs. The size appears to be approximately 2'6"x4'.
At the bottom of the stairs going to the second level of the home, Rudy stands in front of what appears to be either an Isfahan design or Kashan design rug. Isfahans and Kashans are very popular for their use of red, white and blue coloring. Isfahans tend to have ivory white backgrounds more often than kashan rugs, however both often feature a centered medallion with an oval shaped field cupped by each dark blue spandrel as seen above. The size of this rug is probably a 3'x5' or so.
With most certainty, the rug seen in the living room/tv area would be considered an "antique Tabriz design." One indicator of this would be the ornate yet delicate vines as seen in the field and border of top and bottom photograph, as well as the shape and heavily contrasted design pattern of black/navy centered on an ivory medallion. The spandrels in each corner of the field also have a three leaf shape as seen in the bottom photograph under Cliff's knee, which helps to eliminate other possible rugs which this design could be attributed to.
The orange arrow as seen in the image captured below points to a rug measuring approximately 4'x6'. Going by the coloration, shape of the medallion, open field design, and general look of the rug, we believe this to be what would be considered a Kerman Design.