You haven’t described what exactly you do in the part, but from what I see it doesn’t look like a good system design.
It’s absolutely not a good idea to propagate POST requests to ALL components on the page, simply because you never know which of them need it. If you want a part to perform some business logic which will be triggered on POST, then you should post directly to this part.
If you are posting to the page then you obviously have to handle the POST inside the page to let the page control what to do with the data it received - hence the term “page controller”.
Not sure why you mention duplication of the code. If your page controller needs to execute some business logic stored in the part controller, then you simply export function inside the part controller and then call it from the page controller. Or even have a separate helper class where you store business logic which can be called from anywhere.
/parts/part1/part1.js:
exports.hello = function(req) {
...
}
/parts/part2/part2.js:
exports.hello = function(req) {
...
}
/pages/default/default.js:
var part1 = require('../../parts/part1');
var part2 = require('../../parts/part2');
export.post = function(req) {
part1.hello(req):
part2.hello(req):
}