Think you need to find the paperwork that relate to your house purchase first - your solicitor should have asked the local authority for a 'plan' of the property including the boundaries. Then check with the Land Registry or your paper deeds for the boundaries and any underground structures.
Worse case go and see your local highways people for their view on the subject.
You may find that you own or have rights to the land up to the path, road side (i.e. Kerb) or even to the centre of the road.
Where we live the road outside half the front was unadopted on initial purchase. According to the county records office we owned to the path edge on the adopted highway and to the Kerb on the (originally) unadopted highway. Like you we became 'fed-up' wiith people parking on our front lawn so we put large rocks on the lawn to stop parking, that stopped the parking until a neighbouring busy-body reported the rocks to the council. First we knew about it was when we had a letter from the county highways telling us that the rocks were on council ground and remove them immediately and restore the grass or face prosecution. That letter was companied by a drawing that looked like it had been drawn by a five year old 'stating' where our boundaries are and what was, in their eyes, council land.
After a number of postal exchanges I went with my copies of deeds and the drawing supplied by the county council to the highways people. As our drawing from the time of purchase had been issued by the county records and was more than 10 years old the highways had difficulty in making us do much more than move the rocks.
Council do like to claim land but expect the locals to maintain such, from your photo's it does look like the land was possibly part of your property environs originally - particularly as you have a window looking onto the ground.
