Mixed-parity module emerges for instance when a de Rham Galois representation is being tensored with a square root of cyclotomic character, which produce half odd integers as the corresponding Hodge-Tate weights. We build the whole foundation on the $p$-adic Hodge theory in this setting over small $v$-stacks after Scholze and we also consider certain moduli $v$-stack which parametrizes families of mixed-parity Hodge modules. Examples of the small $v$-stacks in our mind are rigid analytic spaces over $p$-adic fields and moduli $v$-stack of vector bundles over Fargues-Fontaine curves.
