Edit in JSFiddle

var json = {
  "elements": [
    {
      "type": "radiogroup",
      "hasOther": true,
      "isRequired": true,
      "name": "favoritePet",
      "title": "What is your favorite pet ![A parrot](https://surveyjs.io/Content/Images/examples/markdown/image_16x16.svg =16x16) ?",
      "choices": [
        {
          "value": "dog",
          "text": "Dog: ![A dog](https://surveyjs.io/Content/Images/examples/markdown/dog.svg =14x14)"
        }, {
          "value": "cat",
          "text": "Cat: ![A cat](https://surveyjs.io/Content/Images/examples/markdown/cat.svg =14x14)"
        }, {
          "value": "parrot",
          "text": "Parrot ![A parrot](https://surveyjs.io/Content/Images/examples/markdown/parrot.svg =14x14)"
        }
      ]
    }
  ],
  completedHtml: "Your favorite pet is <b>{favoritePet}</b>."
};

Survey.StylesManager.applyTheme("defaultV2");

var survey = new Survey.Model(json);

survey.onComplete.add(function(result) {
	document.querySelector('#result').innerHTML = "result: " + JSON.stringify(result.data);
});

var converter = new showdown.Converter();
survey.onTextMarkdown.add(function(survey, options){
    //convert the mardown text to html
    var str = converter.makeHtml(options.text);
    //remove root paragraphs <p></p>
    str = str.substring(3);
    str = str.substring(0, str.length - 4);
    //set html
    options.html = str;
});

$("#surveyElement").Survey({
  model: survey
});

<div id="surveyElement"></div>

<div id="result"></div>