The geology of Bangalore is such that bore wells can be drilled by using Down-The-Hammer (DTH) rigs requiring several tens of feet space for up-and-down vertical movement of the drilling shaft. The space of 10 feet available between the ground level and the roof of the parking area is too small for taking up drilling.
If one can manage and face the associated problems, one can take up construction of the bore well in the neighbour's house where a well could be drilled or in the road poramboku abutting his/her house.
If it’s inevitable to drill the borewell at the location you have mentioned, then a tripod based drilling option can usually be adopted (Used for drilling in narrow lanes in the cities). The height of the hoist is relatively smaller than the hoist mounted on the drilling rigs. If this is around 11 or 12 feet, then excavate the area you are planning to drill the borewell in (say 6x6 feet size) to a required depth to accommodate the height of the moist. However, water coming out of the bore during drilling needs to be diverted manually or using a diversion drain. Contact a good driller who can study the feasibility of this option.