.eslintrc.js

parserOptions: {
  parser: 'babel-eslint',
  sourceType: 'module',
  allowImportExportEverywhere: true
}

应为:

parser: 'babel-eslint',
parserOptions: {
  sourceType: 'module',
  allowImportExportEverywhere: true
}

来源: https://eslint.org/docs/user-guide/配置#specifying-parser

使用(@ vue/cli) .eslintrc.json

{
  "parser": "vue-eslint-parser",
  "parserOptions": {
    "parser": "babel-eslint",
    "ecmaVersion": 8,
    "sourceType": "module"
  }
}