Parry and Graham Kelly, his Football Association counterpart, had flown to Geneva yesterday to support the two London clubs in their appeal against the ban - ruling them out of European competition the first time they qualified in the next five years - imposed for their half-hearted participation in last summer's Intertoto Cup.
That appeal, accompanied by a wealth of documentary evidence, was successful, with fines of pounds 90,000 for Spurs and pounds 60,000 for Wimbledon levied instead. The fines, which with the costs add up to around pounds 180,000, will be shared equally between the 20 Premiership clubs.
Parry indicated that the next step would be to try to regain the Uefa Cup place taken away from England because of the attitude of Spurs, Wimbledon and Sheffield Wednesday to the much-derided competition.
"There isn't an obvious route of appeal because this was a decision made by Uefa's executive committee, but we're going to ask them gracefully if they will reconsider," he said. "I don't see this as a kind of victory," Alan Sugar, the Tottenham chairman, said. "All we have done is get back to the position that 90 other clubs in England have naturally every year, the right to earn a place in Europe."
For Tottenham, the value of a successful run in Europe next season could be in excess of pounds 4m, dwarfing their share of the fine, but Sugar stressed that his club had still paid the price for doing the decent thing.
"The reason we entered was to protect British football from the punitive measures of banning all our clubs from Europe that could have devastated the game in this country," he said. "I feel it's a bit like us being accused of robbing a bank when we hadn't, being found guilty, but then being told not to worry because the police would pay the fine."
For Wimbledon, there was a degree of disappointment that they had not been cleared totally and Ned Hammam, brother of the club owner, Sam, had another metaphor. "It's like putting out a fire that we didn't start," he said.
Asked if English clubs would take part in the tournament in future, Kelly replied: "You never say never. You don't know what the situation will be in the future."
Parry added: "We've already decided that we won't take part this summer because of the European Championship. We have learned our lesson now, though. If we're going to take part, we will do so properly - we won't get it wrong again."