Make Polylang compatible with the WP-API. Query language specific WP-API posts via a parameter and add the post language as a new REST field.

functions.php

<?php
/**
 * https://developer.wordpress.org/reference/hooks/rest_this-post_type_query
 *
 * Query language specific posts via "lang" parameter: /wp-json/wp/v2/posts?lang=en
 */

function my_theme_filter_rest_post_query( $args, $request ) {
	$lang_parameter = $request->get_param('lang');

	if ( isset( $lang_parameter ) ) {
		$args['lang'] = $lang_parameter; // https://polylang.pro/doc/developpers-how-to/#query
	}

	return $args;
}
add_filter( 'rest_post_query', 'my_theme_filter_rest_post_query', 10, 2 );
add_filter( 'rest_page_query', 'my_theme_filter_rest_post_query', 10, 2 );
//add_filter( 'rest_{my_custom_posttype}_query', 'my_theme_filter_rest_post_query', 10, 2 ); // Custom posttype


/**
 * https://developer.wordpress.org/rest-api/extending-the-rest-api/adding-custom-endpoints
 *
 * Register a new REST field "language" and add it to the post data
 * 
 */

add_action( 'rest_api_init', function () {

	register_rest_field( 'post', 'language', my_theme_register_postlanguage_function() );
	register_rest_field( 'page', 'language', my_theme_register_postlanguage_function() );
	//register_rest_field( '{my_custom_posttype}', 'language', my_theme_register_postlanguage_function() ); // Optional: Custom posttype

});

function my_theme_register_postlanguage_function() {
	return array(
		'methods'         => 'GET',
		'get_callback'    => 'my_theme_get_postlanguage_function',
		'schema'          => null,
	);
}

function my_theme_get_postlanguage_function( $data ) {
	$post_id = $data['id'];

	return ( function_exists( 'pll_get_post_language' ) ? pll_get_post_language( $post_id ) : null );
}

Simple codes for all post types

add_action('rest_api_init', function () {
    foreach (get_post_types(array('show_in_rest' => true), 'objects') as $post_type) {
        add_filter('rest_' . $post_type->name . '_query', 'wp_rest_filter_add_filter_param', 10, 2);
    }
});

function wp_rest_filter_add_filter_param($args, $request)
{
    $lang = $request->get_param('lang');


    if (!$lang) {
        return $args;
    }

    $args['lang'] = $lang;

    return $args;
}

@everdha thanks.
@mustafakucuk your code didn't work in functions.php

have a good day