Let's try to write out a list.
(Leaving out dims on purpose - no help here!).
In key C major - C=I, Dm=ii, Em=iii, F=IV, G
=V, Am=vi.
In key Am - Am=i, C=III, Dm=iv, Em=
v (E=V),
F=VI, G=VII.
In key Cm - Cm=i, E&flat;=III, Fm=iv, Gm=v (G=V), A&flat;=VI, B♭=VII.
There is NO natural minor key .
There are natural minor notes, but only minor keys!
Does the list above hold your answer?
Cont...
In key C major, going to key C minor, chord 3 is E&flat;.
So, using C major notation, it'll be ♭Ⅲ .
Chord 6 is A&flat;, so it'll be ♭Ⅵ .
music.stackexchange.com/users/2808/Tim
3 major to 6 parallel minor is correct, iii-bVI.
3 minor to 6 parallel major is bIII-vi.
One should establish the correct interval from the tonic when labeling chords with Roman numbers regardless of whether it’s a major or minor key.
In the key of C (major or minor) the first example is Em-Ab and the second is Eb-Am Also, FYI chords are not generally called “the third chord” but rather “the three chord” or in the case of your first example, “the three minor chord to the flat six chord”.
If a chord is anything other than a major triad it should be identified.
music.stackexchange.com/users/67196/John Belzaguy
If I understand the question (correct me if I misunderstood), for example in Cm, you wish to go from and E major to an A major chord.
The way I've seen this notated would be #III followed by #VI.
The # before the chord means raise the root and the numerals mean a major chord.
music.stackexchange.com/users/23838/ttw
Let’s say we’re in a-minor.
You mean: III(a-minor) =
C -> bVI of parallel key (A-major) =
F your question: iii (A-major) =
c#m ->
bVI of parallel key (a-minor) =
F
your question: Is this bVI F-major or f-minor chord?
As you change from A-major to a minor bVI = f,a,c = bVI -> F-major chord.
music.stackexchange.com/users/56219/Albrecht Hügli
This question illustrates perfectly the sort of muddle that is caused by labelling the three triad of a minor scale III rather than &flat;III.
A minor 3rd is still a minor 3rd whatever scale it occurs in.
It's &flat;III in my book.
However, people DO use the convention of calling Eb &flat;III in a C major context, III in a C minor context.
All I can advise is to make it clear which convention you're using.
Illustrating a discussion with notated examples can help.
(Also, is there some confusion between 'parallel major' and 'relative major'?)
music.stackexchange.com/users/17003/Laurence Payne
If I'm in a major key and I have a progression from the three chord to the six chord borrowed from the parallel minor (modal mixture) I would go:  iii bVI     But if I was in a natural minor  key  scale and I want my progression to go from the three chord to the six chord borrowed from the parallel major then would it be:  bIII vi  or would it be:  III vi  <-- because isn't the three chord in a minor scale generally considered III?
How to notate a borrowed chord if you're in a key other than major?
music.stackexchange.com
<roman-numerals>
music.stackexchange.com/questions/98074
music.stackexchange.com/users/None/
This question illustrates perfectly the sort of muddle that is caused by labelling the three triad of a minor scale III rather than &flat;III. A minor 3rd is still a minor 3rd whatever scale it occurs in.  All I can advise is to make it clear which convention you're using.    Make it clear which convention you're using, and Illustrating a discussion with notated examples can help. 
Minor 3rd is still a minor 3rd whatever scale it occurs in. People use the convention of calling Eb &flat;III in a C major context. Make it clear which convention you're using, and Illustrating a discussion with notated examples can help. 