Right. The official answer is that in the early days, as in the 'Bushy Park Time Trial' and then the its first offspring, the 'Wimbledon Common Time Trial' club vests were discouraged.
The one element of this that has been picked up in the thread was that the Time Trial & parkrun was meant to be inclusive. Now this isn't the same as saying running clubs are not inclusive... but the thought was that whippet looking runners sporting club vests MAY put off new runners who may feel that this wasn't an event for them. This though was not the primary reasons.
The event was a 'time trial' for individuals, not a race. As such seeing as runners were not representing clubs, but only themselves, club vests were not required.
This was actually quite important as before the parkrun concept spread with backers cash (Nike, Lucozade) etc. The events were were organized without any insurance which would have been required if they had been billed as races. As such the less race like appearance the better. So lots of runners wearing all sorts of tops looks a lot less race like than a field made up of 50% club vest wearers! If you think this sounds crazy - just remember the reaction of that old chap in Newcastle who tried to get the parkrun there banned by the AAA!
(Some local clubs in London also objected to the Time Trial/parkrun concept - but as you all know they soon saw the benefits instead & very came on board. Helping volunteer at them, hand out fliers for local races & build bridges with people who normally would never have thought that a running club was for them).
As the concept of parkruns has grown, so the need to constantly remind officials/land owners/councils etc that these aren't races (and thus chargeable for using facilities, having to have St Johns in attendance etc), has diminished so has the original 'no club vests' ethos.
I'd say that now the 'no club vest mindset' is very much an old school mentality kept up only by the early adoptees. With every new event this ethos is considered neither here nor there, & no one really cares a jot. Even in Wimbledon where Ian sports his TH&H vest with pride come rain or shine. As people involved in parkrun only care that people are out running. Failing that just run bare chested - it worked for Craig Mottram & he's still the fastest ever around a parkrun course.