Yes, your parents can visit with you in a university abroad, as long as they meet the visa requirements of the country where you are studying. They will need to apply for a visitor visa, which will typically allow them to stay for 6 months or 1 year. Until and unless student don’t settle down, parents can stay with them.
As far as Travel in Holidays or stay Abroad is concerned so it is totally depend on you. University allows students to go back their homes in Holidays. Many students go their homes and few students stay in abroad only. They get involved in activities and events on campus, Travel within the country where they studying and preparing for exam or doing extra studies.
Consider your budget, your interests, and the terms of your student visa when making your decision.