Is it disobedience towards his parents if he travels to seek provision and leaves them?

Firstly: 

If one or both of your parents needs you to serve them, and
there is no one else who can do that after you travel, then you should not
travel without their permission. 

If they give you permission to travel or they do not need
you, either because they are able to take care of themselves or because
there is someone else who can look after them, then there is nothing wrong
with you travelling in order to get married in that case, so as to preserve
your chastity. But even if they do not give you permission, your travelling
is not regarded as disobedience towards them. 

But it is undoubtedly better to please them and to explain to
them that travelling is in your best interests, and that you will never
neglect them, and that you will be away only so long as you need to, then
you will come back to them, and so on. 

The scholars (may Allaah have mercy on them) stated that it
is permissible for a son to travel in order to seek provision without his
parents’ permission, subject to the condition that travelling is safe and
will not put the son in any danger, and that they do not need him to be
there. 

Al-Kaasaani said in Badaa’i’ al-Sanaa’i’ (7/98), in
his discussion on jihad if it is fard kifaayah (a communal obligation): 

It is not permissible for a slave to go out except with the
permission of his master, or for a woman to go out except with the
permission of her husband, because serving one’s master and fulfilling the
husband’s rights are individual obligations (fard ‘ayn) which take
precedence over communal obligations (fard kifaayah). Similarly a son should
not go out without the permission of his parents, or of one of them if the
other is dead, because honouring one’s parents is an individual obligation
which takes precedence over communal obligations. 

The basic principle is that if travelling is not safe and the
traveller may die or be exposed to severe danger, it is not permissible for
a son to travel without his parents’ permission, because they love their son
and will be harmed by that. If there is little danger involved, it is
permissible for him to go out without their permission so long as he does
not neglect them, because there is no harm involved. Some of our shaykhs
granted a concession allowing a son to travel in order to seek knowledge
without parental permission, because they will not be harmed by that, rather
they will benefit from it, so it cannot be described as disobedience to
parents. End quote. 

Al-Sarkhasi said in al-Siyar al-Kabeer (1/197): 

If a person wishes to undertake a journey that is not for
jihad – such as for business, Hajj or ‘Umrah – and his parents object to
that, but there is no fear that they will suffer hardship because of his
absence, then there is nothing wrong with him travelling, because in most
cases these journeys are safe and his travelling will not cause them undue
hardship. The sorrow at his absence is offset by their looking forward to
his return. But if there is the fear of danger on the journey, then it comes
under the same ruling as going out for jihad, because it is more likely that
he will die on such a journey. End quote. 

Al-Nawawi said in al-Majmoo’ (8/314): 

If a son wants to travel in order to seek knowledge, then
al-Musannif (i.e., Abu Ishaaq al-Shiraazi (may Allaah have mercy on him))
stated at the beginning of Kitaab al-Siyar that it is permissible for
him to do so without his parents’ permission. He said: The same applies to
travelling for business purposes, because in most cases such journeys are
safe. 

Secondly: 

With regard to your not living with them after you get
married, even though they want you to stay with them, there is no sin on you
in that, in sha Allaah, because that does not mean that you are neglecting
them or harming them, especially since there are reasons for you not to do
so, such as the house being small, or your wife wanting to live in her own
home – which is her right – and other such reasons. 

It should be noted that your living apart from them does not
mean that you should not ask about them, or meet their needs or honour them.
Rather these are duties which you owe them, especially now that you are an
adult. Allaah says (interpretation of the meaning): 

“And your Lord has decreed that you worship none but Him.
And that you be dutiful to your parents. If one of them or both of them
attain old age in your life, say not to them a word of disrespect, nor shout
at them but address them in terms of honour.

24. And lower unto them
the wing of submission and humility through mercy, and say: ‘My Lord! Bestow
on them Your Mercy as they did bring me up when I was young’”

[al-Isra’ 17:23, 24]

Honouring one’s parents opens the door to Paradise, so strive
to please them and treat them kindly, and do not upset them. 

And Allaah knows best.