We use constructions like that in recipes. Cocktail recipes especially, these days, but really any instruction on how to make something out of other things that mix together.
A recipe for a screwdriver (vodka with orange juice) might specify "one part vodka to four parts orange juice", meaning that there is four times as much orange juice as vodka. A vinaigrette recipe might say "one part vinegar to three parts oil", meaning three times as much oil as vinegar. It doesn't matter how big a "part" is; it could be 50ml or 3 litres. In other words, it expresses a ratio. "A parts X to B parts Y" means that X and Y are mixed in ratio A:B.
So, in your quotation, it says that his "comfortable English soul" is composed of an equal mixture (1:1 ratio) of irony and convention. This is to some extent invoking a stereotype. The 'soul' in this case is about deep personality traits, rather than anything religious or mystical. Where it invokes convention, it is saying that the person likes to be conventional - to fulfil the general expectations of society. Irony is, I suspect, referring to a tendency to be ironic or see the irony in situations. Both are characteristics stereotypically associated with English people, especially men.
You might use the same construction to suggest that someone's soul or personality was "one part flippancy to one part irreverence" to suggest that they were lighthearted and didn't take things seriously, or "one part seriousness to one part jocularity" to suggest that they were serious but tended to be jokey, showing a contrast between aspects of their character. The quoted example is that, deep down, the person likes to conform to expectation and tends to see or express irony.
