A very little examination of the original drawing will have shown the reader that, as he will have at first read the conditions, the puzzle is quite impossible of solution. 
We have therefore to look for some loophole in the actual conditions as they were worded. 
If the Parson could get round the source of the river, he could then cross every bridge once and once only on his way to church, as shown in the annexed illustration. 
That this was not prohibited we shall soon find. 
Though the plan showed all the bridges in his parish, it only showed "part of" the parish itself. 
It is not stated that the river did not take its rise in the parish, and since it leads to the only possible solution, we must assume that it did. 
The answer would be, therefore, as shown. 
It should be noted that we are clearly prevented from considering the possibility of getting round the mouth of the river, because we are told it "joined the sea some hundred miles to the south," while no parish ever extended a hundred miles!