In traditional game semantics, games are \emph{sequential}, \emph{i.e.}, at each step, either Player or Opponent moves (\emph{global polarization}), 
moreover, only a \emph{single} move can be performed at each step.
More recently, \emph{concurrent games} have been introduced, where  global polarization is abandoned, and multiple moves are allowed.
In this paper, we introduce \emph{polarized multigames}, which are situated half-way between traditional sequential game semantics and concurrent 
game semantics: global polarization is still present, however \emph{multiple} moves are possible at each step, \emph{i.e.} a team of Players/Opponents 
moves in parallel. Usual game constructions can be naturally extended to multigames, which can be endowed with a structure of a monoidal closed category
together with an exponential comonad. 
Multigames are useful to model languages with parallel features, \emph{e.g.} they provide an \emph{universal} model of unary PCF with \emph{parallel or}.
Interestingly, the category of polarized multigames turns out to be equivalent to a category of AJM-games with  a new notion of tensor product, where 
at each step the current player performs a move in \emph{at least} one component of the tensor game. This notion of \emph{parallel tensor product}
is inspired by Conway's \emph{selective sum}.