user8201: Is the number of legal chess positions odd or even? Two positions are not the same if they differ in castling rights (i.e. whether K or R have actually moved) or en passant capability (i.e. whether the move can actually be made) or who has the move.
The number of times position has been repeated is not part of the definition of position for obvious reasons; nor is the number of moves since last capture or pawn move.
I don’t have the answer myself. This is hard but doable with a bit of programming I think.
EDIT: The answer so far and the comments are clearly heading along the right lines, although not always correct yet: terrific work. I want to give multiple +1s! I will summarize the key points to help focus the work. Pairing mirrored positions is essential i.e. those which allow triangulation. I term positions which have no mirror image “vampires” :-) Any vampire (except for the starting position) must be immediate offspring of a vampire, so it’s maybe easiest to just focus on them.
Someone touched on castling, which is very important. We can partition the vampires into 16 different clans. The head of each clan is an "initial game array" position (i.e. with 32 pieces on apparently original squares) but with specific castling rights disabled (by some initial dance of rooks and knight taking 2n.0 moves).
In a clan, if a castling right remains, then that implies that the rook's pawn can safely capture off the file, the f-pawn can move forward a single square or capture or (on the kingside) the bishop can be captured, without allowing triangulation. If the right is lost then the rook's pawn can only move forward a single square, and the bishops cannot be captured.
En passant is less important but amazingly there are vampire en passant positions. For example:
r1bqkb1r/2pppppp/8/1pP5/8/8/PPPPP1PP/R1BQKB1R w KQkq b6 0 12

where White retains at least one castling right, Black has at least queenside castling rights and White can capture e.p.

user20694: Odd if you count the initial position and Even if not.
Mirrored positions
Almost any position, including checkmates and stalemates can be flipped by switching colors.
Mirroring the chess board horizontally between 4 and 5 rows, you cannot find a position that colors can be switched and produce an illegal position. So for each position there exists a mirrored position, except the start position.
The challenge would be to find a position that cannot be mirrored.
Non Mirrored positions / Corresponding positions
There can exist positions where it cannot be mirrored due to the "turn that cannot be switched". For its this position, there would be a "corresponding position" that also cannot be mirrored.
e.g. 1. Nf3 Nf6 2. Ng1 Nd5 3. Nf3 Nf4 4. Ng1 Nd3+
its corresponding condition is:

Nf3 Nf6 2. Nd4 Ng8 3. Nf5 Nf6 4. Nd6+

Both cannot be mirrored.
Triangulation
Due to Triangulation, all positions after a pawn is pushed can be mirrored. Non mirrored positions can be constructed only with knight moves, moving only knights cannot produce Triangulation. 
Start position
The start position, before any moves, the turn cannot be switched so that black makes the first move.
Calculating positions from knight moves
To solve this for sure, we need to calculate the total positions that arise from only knight moves.

8201: <p>Is the number of legal chess positions odd or even? Two positions are not the same if they differ in castling rights (i.e. whether K or R have actually moved) or en passant capability (i.e. whether the move can actually be made) or who has the move.</p>
<p>The number of times position has been repeated is not part of the definition of position for obvious reasons; nor is the number of moves since last capture or pawn move.</p>
<p>I don’t have the answer myself. This is hard but doable with a bit of programming I think.</p>
<p>EDIT: The answer so far and the comments are clearly heading along the right lines, although not always correct yet: terrific work. I want to give multiple +1s! I will summarize the key points to help focus the work. Pairing mirrored positions is essential i.e. those which allow triangulation. I term positions which have no mirror image “vampires” :-) Any vampire (except for the starting position) must be immediate offspring of a vampire, so it’s maybe easiest to just focus on them.</p>
<p>Someone touched on castling, which is very important. We can partition the vampires into 16 different clans. The head of each clan is an &quot;initial game array&quot; position (i.e. with 32 pieces on apparently original squares) but with specific castling rights disabled (by some initial dance of rooks and knight taking 2n.0 moves).</p>
<p>In a clan, if a castling right remains, then that implies that the rook's pawn can safely capture off the file, the f-pawn can move forward a single square or capture or (on the kingside) the bishop can be captured, without allowing triangulation. If the right is lost then the rook's pawn can only move forward a single square, and the bishops cannot be captured.</p>
<p>En passant is less important but amazingly there are vampire en passant positions. For example:</p>
<pre><code>r1bqkb1r/2pppppp/8/1pP5/8/8/PPPPP1PP/R1BQKB1R w KQkq b6 0 12
</code></pre>
<p>where White retains at least one castling right, Black has at least queenside castling rights and White can capture e.p.</p>
 20694: <p><strong>Odd</strong> if you count the initial position and <strong>Even</strong> if not.</p>

<h2>Mirrored positions</h2>

<p>Almost any position, including checkmates and stalemates can be flipped by switching colors.</p>

<p>Mirroring the chess board horizontally between 4 and 5 rows, <strong>you cannot find a position that colors can be switched and produce an illegal position</strong>. So for each position there exists a mirrored position, except the start position.</p>

<p>The challenge would be to <strong>find a position that cannot be mirrored</strong>.</p>

<h2>Non Mirrored positions / Corresponding positions</h2>

<p>There can exist positions where it cannot be mirrored due to the "turn that cannot be switched". For its this position, there would be a "corresponding position" that also cannot be mirrored.</p>

<p>e.g. 1. Nf3 Nf6 2. Ng1 Nd5 3. Nf3 Nf4 4. Ng1 Nd3+</p>

<p>its corresponding condition is:</p>

<ol>
<li>Nf3 Nf6 2. Nd4 Ng8 3. Nf5 Nf6 4. Nd6+</li>
</ol>

<p>Both cannot be mirrored.</p>

<h2>Triangulation</h2>

<p>Due to <a href="https://en.wikipedia.org/wiki/Triangulation_(chess)" rel="nofollow noreferrer">Triangulation</a>, <strong>all positions after a pawn is pushed can be mirrored</strong>. <strong>Non mirrored positions can be constructed only with knight moves</strong>, moving only knights cannot produce Triangulation. </p>

<h2>Start position</h2>

<p>The start position, before any moves, the turn cannot be switched so that black makes the first move.</p>

<h2>Calculating positions from knight moves</h2>

<p>To solve this for sure, we need to calculate the total positions that arise from only knight moves.</p>
