PHP 8.2 deprecates the dynamic creation of properties on PHP classes (T314099: PHP 8.2: Dynamic property creation is deprecated). While this can be suppressed (and in some cases, may be the simplest short term fix), we should look at moving away from this pattern in the Parser longer term.
It's common for extensions to add their own properties to the parser; T324890: Deprecated: Creation of dynamic property Parser::$extTemplateStylesCache is deprecated in Hooks.php on line 208, T324891: Deprecated: Creation of dynamic property Parser::$scribunto_engine is deprecated and T324901: Deprecated: Creation of dynamic property Parser::$extCite is deprecated in CiteParserTagHooks.php on line 94 for example.
On https://gerrit.wikimedia.org/r/c/mediawiki/core/+/866668 which would add #[AllowDynamicProperties] to Parser, @Tgr suggested a longer term improvement for this:
Maybe Parser could have a mechanism for getting/setting extension properties, like ParserOutput does.