Unfortunately most airport websites don't give stands, at least for arriving aircraft.
Flightstats does, but only for US carriers so not much use for European airports I'm afraid.
UK and Global Airport Departures & Arrivals
For Heathrow try lhr-lgw.com -
Heathrow Movements
If you have excel You can download all movements for the airport for a day in one go. Cut & paste into excel, sort by gate and you will be able to see which aircraft stay on stand and which are towed on/off. Bear in mind that times given are for landing & airborne and that a typical taxi in time is 10 mins & taxi out 20+ mins.
As spannersatcx & BOAC4ME have said and aircraft will probably not stay on stand for more than three hours at peak time. Virtually all short haul aircraft will stay on stand as they generally have short turn arounds but many long haul have much longer turnarounds for scheduling reasons, the example of CX has been given.
In my experience it is possible to depart on time from a remote with a full load - if you are lucky - but it is still a hassle and not popular with passengers. In the days when BA operated into T4 it could actually be quicker to be bused in from a remote if a/c were queuing for stands as often happened.
I know that in BAA days aircraft on stand over a certain period at peak times got charged four times the standard rate and for that matter there was a rebate for bused passengers. I'm afraid I'm sure of the current position now. I do know that the Heathrow peak lasts most of the day these days.