A extractive variation. This approach assumes that a position ('QB', 'RB', etc.) cannot be confused with the second or any subsequent field of a player's name.

c:\@Work\Perl>perl -wMstrict -MData::Dump -le
"my $str = 'QB Carson Palmer RB Chris Ivory RB Eddie Lacy WR A. J. Gre
+en '
         . 'WR John Brown  WR WR Grace  WR Pele TE Billy Bob Bennett '
         . 'FLEX   A.P. Hill   DST   Panthers';
 print qq{[[$str]]};
 ;;
 my $result = {};
 ;;
 my $position = qr{ \b (?: QB | RB | WR | TE | FLEX | DST) \b }xms;
 my $player   = qr{ \S+ (?: \s+ \S+)*? }xms;
 ;;
 while ($str =~ m{ ($position) \s+ ($player) (?= \s+ $position | \z) }
+xmsg) {
   ;;
   my ($posn, $name) = ($1, $2);
   ;;
   push @{$result->{$posn}}, $name;
   ;;
   }
 ;;
 dd $result;
"
[[QB Carson Palmer RB Chris Ivory RB Eddie Lacy WR A. J. Green WR John
+ Brown  WR WR Grace  WR Pele TE Billy Bob Bennett FLEX   A.P. Hill   
+DST   Panthers]]
{
  DST  => ["Panthers"],
  FLEX => ["A.P. Hill"],
  QB   => ["Carson Palmer"],
  RB   => ["Chris Ivory", "Eddie Lacy"],
  TE   => ["Billy Bob Bennett"],
  WR   => ["A. J. Green", "John Brown", "WR Grace", "Pele"],
}
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The  $player regex could be refined to make it more discriminative of human names, but that's always tricky.