The Anglican Covenant is primarily concerned with international relations between provinces rather than what happens in parishes. However there is a clear direction of travel, which would have a profound effect on parish groups and their ethos.
Churchgoers know the difference between sermons which tell you exactly what to believe and do, and sermons which inform and inspire you and help with your own decision-making. Similarly those who attend bible study and discussion groups know the difference between a group where everybody is expected to look for 'the biblical answer' and accept it as 'what all Christians should believe', and a group where people are permitted to think for themselves, express diverse views, listen to the views of others and change their minds. More on the historical background to these very different traditions.
The Covenant would increase the tendency for church leaders to declare that a particular belief is the Anglican position. The reason is that it is committed to the view that same-sex partnerships are contrary to the Anglican 'consensus' and proposes to establish a system in which other beliefs, generated by whatever future controversies there may be, can also be declared contrary to the Anglican position.
We can see how the process is envisaged by observing how it has already been applied in the case of same-sex partnerships. Quite clearly there is no consensus about them; some Anglicans think they are permissible, some that they are not. Nevertheless, despite this blatant divergence of belief, the proponents of the Covenant - from the Windsor Report onwards - have insisted that there is an 'Anglican consensus' on the matter. If we ask how they justify this claim, the answers refer us to the Resolution passed at the 1998 Lambeth Conference. In other words, from the perspective of the Covenant's proponents, it does not matter what ordinary Anglicans in their parishes think: Anglican teaching is to consist of what the bishops decree.
If the Covenant works as intended, therefore, we can anticipate an increasing number of doctrines and norms decreed as 'what Anglicanism teaches'. In a parish setting the main effect will probably be on clergy, but this in turn will influence their congregations. Clergy who agree with the official line will be encouraged to generate public objections about clergy who disagree with it. (Again, we have already seen this in operation over same-sex partnerships). Other clergy, who disagree with the official line or simply prefer a more open-minded approach, will be under greater pressure to avoid telling their congregations what they really think, for fear of reprisals. This in turn will increase the pressure to adapt their teaching ministry in accordance with the official line.
In Roman Catholicism the best known example of this situation is the teaching on contraception. Each parish priest must make up his mind whether to agree with the official condemnation, disagree and say nothing, or disagree openly. Parishioners, likewise, learn how much they can say to each priest. In such a context, what is effectively ruled out is any open and honest discussion of the morality of contraception.