Some 19 million passengers used Heathrow in the first nine months of the year, versus 19.3 million who used the airport in the French capital.
Heathrow said Amsterdam Schiphol and Frankfurt were "close behind".
It said all three rivals had adopted testing regimes as a way of people reducing or avoiding quarantine.
By contrast, Britain had been "too slow to embrace passenger testing" and was "falling behind".
Heathrow reported an 84% fall in passenger numbers for the three months to September as losses for the year to date widened to £1.5bn.
"Already in France and Germany, even Canada and Ireland have moved to testing and this is the way to make sure we can protect jobs in the UK as well as protecting people from coronavirus," Mr Holland Kaye told the BBC.
"The government really need to get on and make this happen before the beginning of December if we are going to save people's jobs."
Transport Secretary Grant Shapps has said he wants to have post-arrivals testing up and running in the UK by 1 December.
This would reduce the amount of time arrivals from high risk destinations had to spend in quarantine from 14 days - seen as a big deterrent to travel - to a week.
But Mr Holland Kaye said the industry still needed a "commitment" it would happen.
He added that the only way to really revive air travel was to bring in widespread pre-departure testing that met internationally agreed standards.
The airport has already begun offering pre-departure testing, but only for passengers travelling to destinations that require it.
He urged the UK government to speed up talks with the US over creating a "pilot air bridge" for such a scheme.
"That is the best way to make sure we are no longer importing Covid and also that people can travel with confidence."
Heathrow, which is already cutting 500 jobs, said it forecast passenger numbers of 22.6 million next year - less than a quarter of 2019 levels.
But while it is losing about £5m a day, it said its finances were solid and it had reserves to tide it over until 2023.
The air travel industry has been hit hard by coronavirus, with airlines such as British Airways and Easyjet slashing thousands of jobs.
The International Air Transport Association, which represents 290 airlines, estimates that air traffic will not return to pre-pandemic levels until at least 2024.
Heathrow says it has been overtaken as Europe's busiest airport for the first time by Paris Charles de Gaulle because of a slump in demand for air travel.