Your parents' advice is not only well-meaning, but dead accurate in this situation. My answer would be to do both. Because of the great demand for people in this field, companies are offering excellent starting pay and are willing to take someone without a degree but with the skills to do the job. Just like all other occupations, as more people enter this field, competition for jobs will get more competitive. When that happens, education becomes very important.
Many people today work full time jobs while pursuing their education.
You may not be able to get your degree as quickly but you can still get your degree. Check with your university about the possibilities of finishing part time, evening or on-line. If those options do not exist at your present university, check with other schools in your area. There are sound arguements in both directions. But in the end, it is your choice.