Does amount of URL parameters affect SEO?

It is not the perceived depth of directory that google factors into rankings, urls with so called deep directories can be indexed and ranked just as shorter ones can. http://www.searchenginejournal.com/url-structure-seo/11801/

Though Google does recommend you keep urls as simple as possible. https://support.google.com/webmasters/answer/76329?hl=en

Is the SEO specialist right? Yes and no (or more appropriately, debateable).

That said, the closer you are to the root domain (in your example example.com) the better.

The lower down in the path you are, the less prominence Google will give to your page.

So in your example:

www.example.com/products/page_name/123

Is definitely the preferred way to do it, giving the user and Google enough information about the path, but keeping it short and simple as possible.

Source: Our internal Google Analytics and Search Expert