Let's consider a generic two-body decay $A \rightarrow B + C$. In the center-of-mass frame (i.e, the rest frame of particle $A$), we know that the four-momentum of particle $A$ can be written as $p^{\mu}_{A} = \left(m_{A} , \vec{0} \right)$.
Now, we can also easy calculate the energy of particles $B$ and $C$ as following:
$$E_{B} = \frac{m_{A}^2 + m_{B}^2}{2m_A}, \qquad E_{C} = \frac{m_{A}^2 + m_{C}^2}{2m_{A}}.$$
Adding $E_{B}$ +and $E_C$, I obtain $$E_{B} + E_{C} = m_{A} + \frac{m_{B}^2 + m_{C}^2}{2m_{A}} \ne E_{A} = m_{A}.$$
Clearly, energy conservation cannot be violated. But where is the error then?