We consider surfaces with parallel mean curvature vector (pmc surfaces) in
C
P
n
×
R
\mathbb {C}P^n\times \mathbb {R}
and
C
H
n
×
R
\mathbb {C}H^n\times \mathbb {R}
, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc
2
2
-spheres of a
5
5
-dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres
S
H
2
⊂
M
¯
2
×
R
S_H^2\subset \bar M^2\times \mathbb {R}
of Hsiang and Pedrosa, where
M
¯
2
\bar M^2
is a complete simply-connected surface with constant curvature. When the ambient space is a cosymplectic space form of product type and its dimension is greater than
5
5
, we prove that an immersed non-minimal non-pseudo-umbilical anti-invariant
2
2
-sphere lies in a product space
M
¯
4
×
R
\bar M^4\times \mathbb {R}
, where
M
¯
4
\bar M^4
is a space form. We also provide a reduction of codimension theorem for the pmc surfaces of a non-flat cosymplectic space form.