PRWeek understands it was Parker's promise that swung the decision for Brunswick, which was pitching against Finsbury and Tulchan Communications.
But although BAA group director of corporate comms Duncan Bonfield admitted Parker's regular participation had been discussed, he said he did not expect him 'to be at every meeting'.
'I would not say that it [Parker's involvement] was the decisive factor,' Bonfield told PRWeek.
BAA owns and operates Heathrow, Gatwick, Stansted, Edinburgh, Glasgow and Aberdeen and Southampton airports.
The group also has stakes in and operates airports elsewhere in the world including Budapest in Hungary and Melbourne in Australia.
Although Ferrovial was yet to make a bid as PRWeek went to press, BAA has reportedly agreed with Brunswick and its investment bank NM Rothschild to argue that a £9-per-share bid would undervalue the company. BAA shares closed up at 779p as PRWeek went to press.
BAA is to argue its shares are undervalued because the market price is based too much on the regulated asset base of its three London airports. Other regulated companies, such as water utilities, trade at higher premiums.
However, BAA is heavily regulated by the Civil Aviation Authority, which every five years reviews BAA's activities and sets a level of return it is allowed to make from its airports.
Anything BAA says to convince shareholders it is worthy of a higher bid could affect a future CAA regulatory settlement.
EasyJet founder Stelios Haji-Ioannou has attacked any merger between Ferrovial and BAA, arguing such a deal would concentrate too much power in the hands of a single company.
Ferrovial, best known in the UK for its stake in the London Underground's PFI contracts, already controls Bristol and Belfast's airports.