We develop a support theory for elementary supergroup schemes, over a field of positive characteristic
p
⩾
3
p\geqslant 3
, starting with a definition of a
π
\pi
-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and
π
\pi
-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra
k
[
t
,
τ
]
/
(
t
p
−
τ
2
)
k[t,\tau ]/(t^p-\tau ^2)
, where
t
t
has even degree and
τ
\tau
has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.