Here is the situation:
We use Flask for a website application development.Also on the website sever, we host a RESTful service. And we use Flask-login for as the authentication tool, for BOTH the web application access and the RESTful service (access the Restful service from browsers).
Later, we find that we need to, also, access the RESTful from client calls (python), so NO session and cookies etc. This gives us a headache regarding the current authentication of the RESTful service. 
On the web, there exist whole bunch of ways to secure the RESTful service from client calls. But it seems no easy way for them to live together with our current Flask-login tool, such that we do not need to change our web application a lot.
So here are the question:
Is there a easy way(framework) so the RESTful services can support multiple authentication methods(protocols) at the same time. Is this even a good practice?
Many thanks!
So, you've officially bumped into one of the most difficult questions in modern web development (in my humble opinion): web authentication.
Here's the theory behind it (I'll answer your question in a moment).
When you're building complicated apps with more than a few users, particularly if you're building apps that have both a website AND an API service, you're always going to bump into authentication issues no matter what you're doing.
The ideal way to solve these problems is to have an independent auth service on your network. Some sort of internal API that EXCLUSIVELY handles user creation, editing, and deletion. There are a number of benefits to doing this:

You have a single authentication source that all of your application components can use: your website can use it to log people in behind the scenes, your API service can use it to authenticate API requests, etc.
You have a single service which can smartly managing user caching -- it's pretty dangerous to implement user caching all over the place (which is what typically happens when you're dealing with multiple authentication methods: you might cache users for the API service, but fail to cache them with the website, stuff like this causes problems).
You have a single service which can be scaled INDEPENDENTLY of your other components. Think about it this way: what piece of application data is accessed more than any other? In most applications, it's the user data. For every request user data will be needed, and this puts a strain on your database / cache / whatever you're doing. Having a single service which manages users makes it a lot nicer for you to scale this part of the application stack easily.

Overall, authentication is really hard.
For the past two years I've been the CTO at OpenCNAM, and we had the same issue (a website and API service). For us to handle authentication properly, we ended up building an internal authentication service like described above, then using Flask-Login to handle authenticating users via the website, and a custom method to authenticate users via the API (just an HTTP call to our auth service).
This worked really well for us, and allowed us to scale from thousands of requests to billions (by isolating each component in our stack, and focusing on user auth as a separate service).
Now, I wouldn't recommend this for apps that are very simple, or apps that don't have many users, because it's more hassle than it's worth.
If you're looking for a third party solution, Stormpath looks pretty promising (just google it).
Anyhow, hope that helps! Good luck.
Flask login together with client authentication methods for RESTful service