This is a restricted version of a game that Tanya Khovanova and Joshua Xiong call *odd CM-Nim* in Cookie Monster Plays Games, *The College Mathematics Journal*, 48:4 (2015) 283-293. In their version, you can take any positive number of elements from any one pile or the same positive number from all three piles (or generally any collection of an odd number of piles). Interestingly, the $\mathscr{P}$ positions (see Mark Wildon's comment) for this odd CM-Nim with $k$ piles are the same as the $\mathscr{P}$ positions for standard Nim with $k$ piles (their Theorem 6).

For your version restricting the number of elements that can be taken in a turn, the analysis is different. With 126 states, it's feasible to work out the $\mathscr{P}$ positions by hand. To get started, the $\mathscr{N}$ positions where the next player can win (i.e., move to $(0,0,0)$) in one move are
$$(1,0,0), (2,0,0), (0,1,0), (0,2,0), (0,3,0), (0,0,1), (0,0,2), (0,0,3), (1,1,1), (2,2,2).$$
One description of $\mathscr{P}$ positions is that every possible move from a $\mathscr{P}$ position goes to an $\mathscr{N}$ position. You can see that every possible move from each of
$$(1,1,0), (1,0,1), (0,1,1), (0,4,0), (0,0,4)$$
goes to a state in the previous list of $\mathscr{N}$ positions, so these five positions join $(0,0,0)$ as $\mathscr{P}$ positions. To start the next round of $\mathscr{N}$ positions,
$$(2,1,0), (1,2,0), (1,3,0), (1,4,0), (1,1,2), (1,1,3), (2,2,1)$$
can go to the $\mathscr{P}$ position $(1,1,0)$ in one move. Etc. (Typically, there are fewer $\mathscr{P}$ positions than $\mathscr{N}$ positions.)

Eventually this "backwards induction" will put the initial state $(2,5,6)$ in one of the lists, telling you whether the initial position is a first or second player win. The $\mathscr{P}$ positions also describe optimal play: when it's your turn, move to a $\mathscr{P}$ position if possible.