Is there any security advantage of hiding route parameters
Yes, if they contain anything potentially sensitive.
If you pass information as URL params, either in the query string (?key=value) or as part of the path (/someValue) then these are cached in various places in the browser and host machine.
Most restful services expect you to pass some form of identifier in the URL however you must consider the impact should that identifier be exposed.
For example imagine you have an endpoint that accepts username as a url param (domain.com/users/username) and your username can also be an email address. You essentially expose the users email address for that service to place you do not control (browser history etc).
You need to make the call on how sensitive each data point is and decide if you are happy exposing it (weight up risk).
General rule for me is, if that information can be used to leverage anything over the user or as a stepping stone to expose further info, don't expose it in the URL.