Does he have to ask his parents for permission to travel a short distance for two days?

If one or both parents does not need the son to serve them, nurse them and so on, then there is nothing wrong with him travelling to a safe place where there is no danger without asking their permission, and he is not regarded as disobedient towards them because of that. If he tells them about his trip out of respect for them and to make them feel good, that is better. 

As for a trip in which there is the risk of dying, such as a trip for the purpose of jihad, it is essential to seek their permission so long as jihad is not an individual obligation in his case. 

Al-Kasaani (may Allah have mercy on him) said in Badaa’i’ al-Sanaa’i’ (7/98): He should not go out for jihad except with the permission of his parents or of one of them, if the other is deceased, because honouring one’s parents is an individual obligation so it takes precedence over a communal obligation. 

The basic principle is that with regard to any journey in which there is the risk of dying and where danger is great, it is not permissible for a son to set out without the permission of his parents, because they love their son and will be harmed by that. If there is no great danger involved in a journey, then it is permissible for him to set out without their permission so long as that will not affect them, because it does not cause them any actual harm. Some of our shaykhs granted a concession allowing a son to travel in order to seek knowledge without their permission, because they will not be harmed by that, rather they will benefit from it, so he is not regarded as being disobedient to them. End quote. 

Al-Sarkhasi (may Allah have mercy on him) said in Sharh al-Siyar al-Kabeer (1/197):  

Every journey that a man wishes to undertake except for jihad, such as travelling for business, Hajj or ‘Umrah, if his parents object to that but he does not fear that they will suffer any hardship as a result, then there is nothing wrong with him travelling, because in most cases such journeys are safe and his travelling does not cause them any great hardship. Even though they may miss him due to his absence, this may be tempered by the hope that he will return safely. But if there is the fear that he may be harmed during this journey, then it comes under the same ruling as travelling for the purpose of jihad, because there is clear danger of him dying. If a person wants to travel for the sake of learning, and the route is safe and the place to which he wants to go is safe, then it is no different than travelling for the purpose of business, rather it is more important, because Allah says (interpretation of the meaning): 

“Of every troop of them, a party only should go forth, that they (who are left behind) may get instructions in (Islamic) religion”

[al-Tawbah 9:122]

So there is nothing wrong with him going out for that purpose even if his parents object, so long as there is no fear that they will be harmed by that. End quote. 

Something similar was stated by al-Nawawi (may Allah have mercy on him) in al-Majmoo’ (8/314). 

In conclusion: there is no sin on you for not asking permission from your parents for this trip, because in most cases it is safe, and it seems that they do not need you to serve them or for any other reason, because you live in a city other than theirs. 

And Allah knows best.