Central defender Allan, who broke into the first team last season and started in the penultimate away game at Northampton, has played the full 90 minutes in the last two matches against Hartlepool and Bristol Rovers.
Fellow 18-year-old Davies, a left-back, has started in all York's league games since arriving on loan from Preston in the summer and has had his initial month-long stint extended until January, while midfielder Platt, 19, has made five starts for the Minstermen this season.
"They are three young lads, but they are very good players and I like to give young talent an opportunity," Worthington said.
"Ben Davies has been excellent for us and we were delighted to extend his stay, Tom Platt got into the first team last season and has not looked back and Tom Allan has been given his chance in the last two matches and has done very, very well.
"All three players have been given a chance and they've seized it. They're very inexperienced, but have risen to the challenge of first-team football and we're very pleased with their contributions so far."