No.
They can only follow their published admission arrangements, which have to comply with the Admissions Code.
They must not use subjective criteria - even for faith schools, they must not take into account views on your suitability, but only use demonstrable things which are open to anyone - such as membership of the religion, being on the parish roll if specific church named (you do not have to live in the parish to join in this way), and frequency of attendance.
For non faith schools, typical criteria (after SEN, which are pretty much separate to main admissions; and LAC, former LAC who must be given priority) include catchment priority, siblings, teacher's DC, and tie-break distance from school. Or it could for a few schools be a lottery. There might also be some selective places, or feeder school priority, or fair-banding.
Schools do not have to use all the legally possible criteria, but they may not use any others. Having non-compliant admissions policies is exceptionally rare (though mistakes could I suppose happen), not least as it is a grounds for successful appeal by those who would have got a place if the arrangements had complied.
It's all done by computer these days, sorting and prioritising by the published criteria. There's no-one reading applications and deciding who fits