Parse JS SDK Login into Parse Server running in Node.JS = fail

fellow developers.

I'm migrating an app that was using Parse Hosted and the new code will run inside Node.JS (with express). I am having a bit of a hard time figuring out a problem, though: my JavaScript SDK requests always return "Cannot GET /parse/1/login", as if I was requesting an invalid express route. I know it is valid because if I reproduce the call without the right appID it will return the {"error":"unauthorized"} message.

I'm trying the most basic possible version, using Node.JS code from the Parse docs:

var express = require('express');
var ParseServer = require('parse-server').ParseServer;
var app = express();

var api = new ParseServer({
  databaseURI: 'mongodb://localhost:27017/dev', // Connection string for your MongoDB database
  cloud: '/home/myApp/cloud/main.js', // Absolute path to your Cloud Code
  appId: 'myAppId',
  masterKey: 'myMasterKey', // Keep this key secret!
  fileKey: 'optionalFileKey',
  serverURL: 'http://localhost:1337/parse' // Don't forget to change to https if needed
});

// Serve the Parse API on the /parse URL prefix
app.use('/parse', api);

app.listen(1337, function() {
  console.log('parse-server-example running on port 1337.');
});

The Node.JS server runs OK and I can even query data using cURL requests with no problems at all.

My login code is also the most basic possible version, running directly into the browser console in a page that loads the JS SDK:

Parse.User.logIn("user", "pass", {
  success: function(user) {
    console.log('ok', user);
  },
  error: function(user, error) {
    console.log('error', user, error);
  }
});

The return is simply Cannot GET /parse/1/login (it seems to come from Express, but the message changes if I remove the appID, so I know the Parse Server is getting it). It is worth noting that this very same call worked perfectly if I issue it against api.parse.com servers: {"code":101,"error":"invalid login parameters"}

Any ideas?