Can my boyfriend claim me as a dependent if we live together on the road but or physical address is different?

What do you mean when you say your physical address is still in SD?  If you are maintaining a home in SD, then no he could probably not claim you as a dependent on his return unless he is paying over half of your support during the year.

If you lived with your parents and you simply never changed your official address but all of the following apply then yes, he can claim you as a dependent:

• You lived with him for the entire tax year
• You are not a qualifying child of another taxpayer
• He provided over half of your support for the tax year
• You earned less than $4,300 for the year. 

Yes it's my parents address I haven't changed. And yes he pays all of the expenses

Then yes, as long as you have made less than $4,300 he can claim you for the Other Dependent Credit which is a $500 non-refundable credit. 

Ok. Will I need to verify in any way that I have lived with him?

Not when you file.  If for some reason, in the unlikely event, you would be audited it is possible.  This could be proved by mail coming to your home addressed to you or most likely just an actual statement from both of you saying that yes you live there. 

Ok. Thank you.