This is not a circle of 5ths method, but it works. This is an easy way to both figure out the key signatures of modes and also play the parallel modes.
Say you want to play the Ionian mode in any key, what's the key signature? Of course it's the major key signature for the key you're in. So that if you are on the key of C, there are no sharps or flats in Ionian. C-Ionian, D-Dorian, E-Phrygian, F-Lydian, G-Mixolydian, A-Aeolian and B-Locrian are all related to one another in that they share the same key signature. If you're in E-flat Major, you have 3 flats for E-flat Ionian, and so on, and if you're starting on E-flat Ionian, you similarly use the same key signature for all modes on that key, so that for instance F-Dorian will also have 3 flats, G-Phrygian has 3 flats, and so on. It's the same for every key. Confused yet?

Now to transpose the mode (ie., find a parallel mode), here's the steps you take in this method:
1. First determine what mode you want on what key.
2. Then determine what the INTERVAL difference between the mode you're in, and Ionian mode, going upward from Ionian. Such as Phrygian mode is a Major 3rd up from Ionian, Lydian is a Perfect 4th up , etc.
3. Once you know the interval, mirror that interval DOWN from the note that Ionian starts on. The note you end up on is the key signature (major) that you will use.
4. Using this key signature and the starting key in Step 1, play the mode.
Let's try an example:
1. Let's do Lydian mode on the key of B.
2. The interval difference between Ionian and Lydian is a Perfect 4th UP.
3. A perfect 4th DOWN from B is F-sharp, which is our key signature, which is F-sharp Major (6 sharps).
4. Using the key signature of F-sharp Major, play the mode starting on B.
So then the Lydian mode on the key of B is:
B - C# - D# - E# - F# - G# - A# - B
This will work every time provided you know your intervals well -- substituting a minor 3rd for a major 3rd (for instance) will give you a wrong result.
Here's some more resources for you about modes:
https://www.petethomas.co.uk/jazz-modes.htmlhttps://en.wikibooks.org/wiki/Music:Modeshttps://en.wikipedia.org/wiki/Mode_(music)https://www.wisegeek.com/what-are-the-seven-modes-of-music.htmJosh