This choice leads to an implicit integration scheme. Let us focus on the explicit scheme first. It cannot be parallelised, unless you are solving a very large system of coupled ODEs, in which case, the whole system of equations can be solved in parallel. This is generally true of any ODE problem, and any ODE numerical method. They cannot be parallelised by themselves.
Parallelism is possible for large systems of ODEs only.
The Euler scheme may become unstable. To see that consider two possible solutions x n and x ' n at t n that are very close, so that , where is a very small number. In order to assess the stability of the method we need to consider the evolution of with time. Now, let us assume that , where g is a growth factor . This means that if, for example, is positive then Euler method is bound to be unstable, whatever the choice of .