I think it’s very important to emphasise to anyone proposing to visit a church to see parish records that they need to contact the church in advance. I was in a graveyard in Co. Antrim recently when I was approached by a couple from New Zealand who had turned up unannounced on a weekday and were very disappointed to find the church locked, and apparently no way of gaining access that day.
And the parish records are often not kept in the church anyway. I know of one church locally where the older records are in the Presbyterian Historical Society in Belfast, another where the priest keeps them in the parochial house and a third where, though the records are in the church, they are in a fireproof safe to which only a couple of people have access.
With regard to the Church of Ireland fees, a Rector that I know does charge simply for access to the records, as well as for copies. I can confirm that the rate is £12 an hour, though some flexibility is shown about this, and it is often rounded down to £10. Most parish records in Northern Ireland and some adjacent counties have been copied and can be seen free in PRONI, so that’s a way of avoiding that charge, but a few parishes have not been copied and so inspecting them at the church/vicarage is then the only option.
Most Ministers, Priests & Rectors in Ireland get rather more genealogical enquiries than they can cope with. When you phone up to make an appointment you often hear a little intake of breath when you say why you are phoning. They don’t all welcome the enquiries and some have indicated it interferes with what they see as their proper pastoral work. (One Minister showed me a letter in which the writer asked for copies of all birth and marriage entries for a particular surname, since the church records began. You can imagine how much work that might have involved). A parish priest near me once commented that his priority lay with supporting his living parishioners, not the dead ones. I have never heard of anyone being refused permission to access the records but there is sometimes a delay. And I have heard of letters going unanswered. Phone calls, with plenty of advance notice, may prove more effective. Possibly worth bearing in mind when approaching churches.