Why is there no numbering system for more than three parallel runways?
"Because those are rules."

ICAO Annex 14 (volume 1) provides the following guidance:


  5.2.2.4 A runway designation marking shall consist of a two-digit number and on parallel runways shall be supplemented with a letter. On a single runway, dual parallel runways and triple parallel runways the two-digit number shall be the whole number nearest the one-tenth of the magnetic North when viewed from the direction of approach. On four or more parallel runways, one set of adjacent runways shall be numbered to the nearest one-tenth magnetic azimuth and the other set of adjacent runways numbered to the next nearest one-tenth of the magnetic azimuth. When the above rule would give a single digit number, it shall be preceded by a zero.


and


  5.2.2.5 In the case of parallel runways, each runway designation number shall be supplemented by a letter as follows, in the order shown from left to right when viewed from the direction of approach: 
  
  
  for two parallel runways: “L” “R”; 
  for three parallel runways: “L” “C” “R”; 
  for four parallel runways: “L” “R” “L” “R”; 
  for five parallel runways: “L” “C” “R” “L” “R” or “L” “R” “L” “C” “R”; and 
  for six parallel runways: “L” “C” “R” “L” “C” “R”.
  


Presumably, the rules for runway designation were created before anyone had the fantasy to imagine mega airports with 4 or 5 parallel runways like we have today. They have then later been adopted to accommodate such airports, without changing the core principles, since doing so would potentially require thousands of airports worldwide to rename their runways and change associated procedures. Not to mention the changes that would be required to aircraft and ATC software, meteorological reporting formats, training and certification of various personnel and so on. Simply put, the disadvantages of using a slightly flawed system are significantly smaller than the cost of redesigning it completely.