What's the correct word for the priest who is in charge of a parish in Catholic Church? Provost seems to be of some higher rank, and Parson, I guess, refers to Anglican/Protestant Church? Also, should this position always begin with capital letter? So confusing...
It's called parish priest or pastor:
This section concerns the priest who in the Code of Canon Law is referred to by the term parochus, which is some English-speaking countries is rendered as "the parish priest", in others as "the pastor".
The parish priest is the proper pastor of the parish entrusted to him. He exercises the pastoral care of the community entrusted to him under the authority of the diocesan Bishop, whose ministry of Christ he is called to share, so that for this community he may carry out the offices of teaching, sanctifying and ruling with the cooperation of other priests or deacons and with the assistance of lay members of Christ's faithful, in accordance with the law
(—canon 519 of the Code of Canon Law in the English translation by the Canon Law Society of Great Britain and Ireland, assisted by the Canon Law Society of Australia and New Zealand and the Canadian Canon Law Society)
The pastor (parochus) is the proper pastor (pastor) of the parish entrusted to him, exercising the pastoral care of the community committed to him under the authority of the diocesan bishop in whose ministry of Christ he has been called to share, so that for that same community he carries out the functions of teaching, sanctifying, and governing, also with the cooperation of other presbyters or deacons and with the assistance of lay members of the Christian faithful, according to the norm of law
(—canon 519 of the Code of Canon Law in the English translation by the Canon Law Society of America).)
There are many complications in this field: for example, originally a vicar (same etymology as vice-) was an assistant to the curate who had the 'care of souls'; nowadays it is the other way round. Again, in my (Anglican) church the vicar and the parish priest are different people; given this complexity, trying to find 'the correct word' without being more specific is probably pointless.
In Britain and Europe generally, pastor is another term for priest, and as such is never capitalised. Either (or parish priest) could be used in your context.