If I am leasing space from a building owner and the building does not have enough parking spaces to be compliant with ADA Standards, am I obligated to address this or is it solely the responsibility of the building owner?

Both tenants and landlords have obligations to ensure that facilities and operations are ADA compliant.  The lease may indicate how barrier removal responsibilities are shared but it is important to remember that the lease does not relieve either tenant or landlord from their individual obligations to comply with the ADA. 

The ideal time to address this issue is when you are leasing new space or renewing a lease.  A tenant should request that the landlord agree in the lease to modify those areas which are under the exclusive control of the landlord, such as parking, sidewalks, entrances, and rest rooms.  If the lease specifies that the tenant has control over these areas, then the tenant should request that a provision be added to the lease so that the tenant can make those alterations needed to comply with barrier removal requirements.