Hello, I am a solicitor with over 15 years experience. I will try to help you with this.
Your parents have been occupying and using land for over 26 years as if it was their own without any permission being given or rent paid or any issues being raised by anyone about that occupation of the land? There are now new neighbours who who to claim this land which is a 1 meter strip at the bottom of your mother's garden.
If your mother and or father were possession of the land for at least 12 years before 13 October 2003, then they will probably have acquired a right to the land through adverse possession (ie squatter's rights). On the dates you have given this seems to be the case. To get this properly recognised an application will need to be made to the HMLR under the transitional provisions and possibly the new regime as a fall back ( see this HMLR practice guide https://www.gov.uk/government/publications/adverse-possession-of-registered-land/practice-guide-4-adverse-possession-of-registered-land).