Some parallelisms of PG(3, 5) involving a definite type of spread

There are 372 parallelisms of PG(3,5) which have been explicitly constructed and studied before the present paper. They contain all but one of the 21 nonisomorphic spreads of this projective space. It was not known by now if that particular spread can be included in a parallelism. We give a positive answer to this question by constructing parallelisms invariant under a cyclic group of order 8 some of which contain the spread. Useful invariants are calculated for all the obtained 899 new parallelisms of PG(3,5).