BG-MO 2022 Problem 1

A white equilateral triangle $T$ with side length $2022$ is divided into equilateral triangles with side $1$ (cells) by lines parallel to the sides of $T$ . We'll call two cells $\textit{adjacent}$ if they have a common vertex. Ivan colours some of the cells in black. Without knowing which cells are black, Peter chooses a set $S$ of cells and Ivan tells him the parity of the number of black cells in $S$ . After knowing this, Peter is able to determine the parity of the number of $\textit{adjacent}$ cells of different colours. Find all possible cardinalities of $S$ such that this is always possible independent of how Ivan chooses to colour the cells.