  This paper introduces the parastatistics induced by ${\mathbb Z}_2\times
{\mathbb Z}_2$-graded algebras. It accommodates four kinds of particles:
ordinary bosons and three types of parabosons which mutually anticommute when
belonging to different type (so far, in the literature, only parastatistics
induced by ${\mathbb Z}_2\times {\mathbb Z}_2$-graded superalgebras and
producing parafermions have been considered). It is shown how to detect
${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosons in the multi-particle
sector of a quantum model. The difference with respect to a system composed by
ordinary bosons is spotted by measuring some selected observables on certain
given eigenstates. The construction of the multi-particle states is made
through the appropriate braided tensor product. The application of ${\mathbb
Z}_2$- and ${\mathbb Z}_2\times {\mathbb Z}_2$- gradings produces $9$
inequivalent multi-particle Hilbert spaces of a $4\times 4$ matrix oscillator.
The ${\mathbb Z}_2\times {\mathbb Z}_2$-graded parabosonic Hilbert space is one
of them.
