I am not sure if this Boolean system is correct in both reverse and forward direction ?
$\begin{gathered} \left\{ {\begin{array}{*{20}{l}} {a + b + c}& \equiv &1&{{\text{mod}}\;2}&{\;(1)} \\ {b + c + d}& \equiv &1&{{\text{mod}}\;2}&{\;(2)} \end{array}} \right.\; \hfill \\ \Rightarrow \;a + d \equiv 0\;{\text{mod}}\;2\;\;(3) \hfill \\ \end{gathered} $
In fact, the $\implies$ direction is true. Not the other one.
Indeed, converting your expressions into their equivalent in modulo 2 arithmetic, it is true that :
$$\begin{cases}a+b+c &\equiv& 1 & \text{mod} \ 2 & \ (1)\\b+c+d &\equiv& 1& \text{mod} \ 2 & \ (2)\end{cases} \ \implies \ a+d \equiv 0 \ \text{mod} \ 2 \ \ (3)$$
Just add for that (1) and (2), using the rule $2x \equiv 0 \ \text{mod} \ 2$, to get (3).
In the reverse direction: taking $a=d=0$ and $b=c=1$, equation (3) is fulfilled while neither equation (1) nor equation (2) is fulfilled.