How to pass variable as parameter, object type, to Azure DevOps template

To reduce duplicates I want to rewrite our Azure DevOps pipelines. The main pipelines looks like:

parameters:
  - name: MODULE_Foo
    type: boolean
    default: false
  - name: MODULE_Bar
    type: boolean
    default: false
...
  - name: MODULE_X
    type: boolean
    default: false
...rest of parameters
 - name: BRANCH_NAME
    type: string
  - name: CHECKOUT_TAG
    type: boolean
    default: false

extends:
  template: template1.yml
  parameters:
    MODULES: ${{paramaters}}
    BRANCH_NAME: ${{parameters.BRANCH_NAME}}
    ...rest of parameters

The template template1.yml:

parameters:
  - name: MODULES
    type: object
    default: {}
  ...rest of parameters

stages:
  - stage: Build
  jobs:
      - job: BuildAndDeploy
        steps:
          - bash: |
              MODULES=$(echo "${{ convertToJson(parameters.MODULES) }}" \
                | sed -E 's/^([[:space:]]+)/\1"/;s/(:[[:space:]]+)/"\1"/;s/,$/",/;s/([^{},])$/\1"/' \
                | jq 'with_entries( select(.key | startswith("MODULE_") ) )' \
              )
              echo "##vso[task.setvariable variable=MODULES;]$MODULES"

          - template: template2.yml
            parameters:
              MODULES: $(MODULES)
            ...rest of parameters

And finally template2.yml:

parameters:
  - name: MODULES
    type: object
    default: {}
  ...rest of parameters

  - ${{ each module in parameters.MODULES }}:
    - ${{ if and(startsWith(module.Key, 'module'), eq(module.Value, true)) }}:
      - bash: |
          echo "Module to build: ${{module.key}}, Value: ${{module.value}}"

Now I'm stuck.

1. How to pass a variable as parameter to a template? To the template is passed empty string '' for ${{variables.MODULES}} or '$(MODULES)' for $(MODULES). But never generated string. I think that $(MODULES) is valid for this situation.

2. How to create a valid object, which can be iterated inside each loop in template? I think that my construction with convertToJson, sed, and jq is not correct.

My goal is to eliminate the MODULE_ parameter group from the templates. Currently, every template contains a duplicate of this parameter section.