  Within the framework of the Duffin-Kemmer-Petiau (DKP) formalism with a
deformation, an approach to the construction of the path integral
representation in parasuperspace for the Green's function of a spin-1 massive
particle in external Maxwell's field is developed. For this purpose a
connection between the deformed DKP-algebra and an extended system of the
parafermion trilinear commutation relations for the creation and annihilation
operators $a^{\pm}_{k}$ and for an additional operator $a_{0}$ obeying
para-Fermi statistics of order 2 based on the Lie algebra $\mathfrak{so}(2M+2)$
is established. The representation for the operator $a_{0}$ in terms of
generators of the orthogonal group $SO(2M)$ correctly reproducing action of
this operator on the state vectors of Fock space is obtained. An appropriate
system of the parafermion coherent states as functions of para-Grassmann
numbers is introduced. The procedure of the construction of finite-multiplicity
approximation for determination of the path integral in the relevant phase
space is defined through insertion in the kernel of the evolution operator with
respect to para-supertime of resolutions of the identity. In the basis of
parafermion coherent states a matrix element of the contribution linear in
covariant derivative $\hat{D}_{\mu}$ to the time-dependent Hamilton operator
$\hat{\cal H}(\tau)$, is calculated in an explicit form. For this purpose the
matrix elements of the operators $a^{\phantom{2}}_0$, $a_{0}^{2}$, the
commutators $[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0},
a^{\pm}_{n}\hspace{0.02cm}]$, $[\hspace{0.03cm}a^{2}_{0},
a^{\pm}_{n}\hspace{0.02cm}]$, and the product
$\hat{A}\hspace{0.03cm}[\hspace{0.03cm}a^{\phantom{\pm}\!}_{0},
a^{\pm}_{n}\hspace{0.02cm}]$ with $\hat{A}
\equiv\exp\hspace{0.02cm}\bigl(-i\frac{2\pi}{3}\,a_{0}\bigr)$, were preliminary
defined.
