Clarification regarding definition: Two way non deterministic finite state Automata
As you clearly state, an input is accepted, by definition, if one of the parallel branches accepts. This means that an input is accepted if and only if at least one of the parallel branches accepts. So if one brach accepts and another rejects, the input is accepted. 

In non-deterministic automata, acceptance and rejection are not symmetric. An input is accepted iff there is some accepting computation path. An input is rejected iff all computation paths are rejecting.