On balance running with my dogs (at a Parkrun its either Seela the husky, who is nearly 12, or Beans the Springer (4) as you can run with only one) is about neutral in terms of time. Time gained by being towed is cancelled out by time spent drinking from puddles, cattle troughs and other doggy functions.
At Montacute's 1st anniversary Parkrun there was Liz and Martin Yelling. Montactue is a 'proper' park run, two laps around a large field. Martin blew passed most of the runners and finished in about 23 minutes pushing a double buggy with twins.
Canix is OK, I've done the Neolithic 1/2 marathon but most of the events are 5k and the idea of driving for 1-2 hours to get to an event that is over in 30 minutes does not appeal.