Police say they handed out fixed penalty fines of £200 after the pair had ignored several other warnings since the lockdown was introduced.
The pair were also reported by members of the public numerous times for breaking the regulations.
The fines came during a busy 48 hours for Horncastle-based officers at the weekend.
They handed out seven other fines for lockdown breaches - by hare-courses.
One group had travelled to the area from Pembrokeshire in Wales and the other from Bishop Auckland in County Durham.
More fines will follow this week as police revealed they would be ‘knocking on the doors’ of people who attended an illegal house party in Woodhall Spa at the weekend. Officers stress they have followed the national ‘four E guidelines’ of dealing with breaches- engage, educate, encourage and enforce, as a last resort.
However, only last week they warned they would ‘come down hard’ on any blatant breaches as police forces across the country adopted a tougher stance.
With regards to the two elderly people, PCSO Nigel Wass told the News: “The pair had been warned on several occasions to stay at home - unless they had a valid reason to be out and about.
“We also received a number of complaints from members of the public who were worried about their own safety and the safety of the two individuals themselves.
“We received another complaint on Saturday and the pair were found in the launderette eating bags of chips.
“They admitted they weren’t doing their washing.
“Hopefully, the fines will get the message across that people should only leave home for a valid reason.
“Clearly, eating chips in a public, indoor place is not a reason to be out and about.”
Horncastle officers were also called to the Sibsey area to investigate reports of harecoursing.
Three individuals quickly left the scene in a vehicle.
PCSO Wass explained: “They thought they had got away but we are grateful to the Rail Network signalman who shut the level-crossing gates on the line near Sibsey.
“That allowed us to catch up with the vehicle and its occupants who had travelled from Pembrokeshire.
“They were given fixed penalty fines and told to leave the county - and not come back.”
PCSO Wass added the vehicle was later ‘pinged’ for a speeding offence by an APR camera in Peterborough.
Five other individuals received £200 fines after being discovered harecoursing in the Hemingby area.
They had travelled from Bishop Auckland.
Reports from the public also alerted officers to the house party in the Sterling Place area of Woodhall Spa.
PCSO Wass added: “The people who attended thought they had been clever by parking in adjoining streets.
“However, we have names and addresses and they can expect a knock of their front door this week.”
Police say they were ‘astounded’ by the number of brazen Covid breaches.
PCSO Wass added: “It is a very small minority of people who seem to think they can do what they want - without any consequences.
“People are dying from this virus - the NHS is struggling to cope.
“Anyone who breaks the regulations is putting themselves and others in danger.
“Please, stay at home.”
•Police also arrested two people in the Horncastle area at the weekend for what they describe as ‘rural crime’ offences.