As a significant advance towards implementing topological quantum computation, a pressing goal for the community is to successfully demonstrate the functioning of Majorana fermio -based topological qubits. Non-local pairs of such fermions share an electronic state that can be either occupied or empty, making such a pair a parity qubit. Semiconducting nanowires, and more recently, chains of ferromagnetic atoms, in proximity to a superconductor, have received prominent attention for their ability to nucleate Majorana fermion states bound to their ends. As with conventional quantum computing, implementing topological quantum computation in a materials system can be best achieved by investigating multiple routes. Here we present an alternative approach, networks of lateral superconductor-topological insulator-superconductor Josephson junctions as a viable, highly promising candidate that has several advantages for supporting Majorana fermion-based topological qubits. The architecture consists of superconducting islands deposited on a topological insulator substrate forming long Josephson junction pathways. Magnetic fields threading the junction nucleate Majorana fermions localized in the junction whose locations can be controlled by applied currents. We propose protocols essential for topological qubit operations involving: i) tuning the coupling between the Majorana states to affect braiding and non-Abelian rotations, and ii) measurement of non-local parity transitions induced by such a rotation. The proposal makes use of local magnetic field or currents pulses to control the spacing of Josephson vortices that host MFs to perform operations and quantum dots to readout the parity of Majorana fermion pairs. We report our progress in the experimental realization of this architecture.