As an extension of your idea, one could just do $count steps starting from zero, and then map the result to the desired interval.

use strict;
use warnings;

sub parts {
  my ($start, $end, $count) = @_;
  my @steps = (0);
  push @steps, rand()+$steps[-1] for 1..$count;
  my $scale = ($end-$start)/$steps[-1];
  $_ *= $scale, $_ += $start for @steps;    
  return @steps;
}

my @parts = parts 0, 10, 10;
print "@parts\n";
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This, of course, is not able to generate skewed distributions, so less flexibility.