I fully agree with the answer from @Ash. But, let me add, on an Interpersonal note...
The concept "Yes, and..." is borrowed from group improv stage play, such as the show "Whose Line Is It Anyway?" The concept is that each action or line in the performance must always "agree" with the actor before, then add more to the scene; never level an agreement; never reject a line or action. (See notes at end.) When I discussed this with a friend from Calvin College on their acting team, he summarized it as "Yes, and..."
While intended for improv plays, I have found great success with "Yes, and..." in any uncharted and "unscripted" circumstances in my own life. When changes happen, I agree and move forward with some kind of contribution. It revolutionized my own relationships, especially positively breaking the ice with people I don't know. I mastered the skill in cold-contacting and door-to-door sales.
This is a kind of cold-contact situation with a high likelihood of getting a "cold shoulder" in return. Asking to mow the lawn is quite unusual and would need a lot to succeed. You can't just ask for the past while worried; you must try to make some small improvement while not being overwhelming.
The need for adding value all the time also goes back to Dr. Deming and his "Continuous Improvement" paradigm he taught to Japan. In that, we're either growing or dying; there is not "maintaining status quo". (Deming Cycle: The Wheel of Continuous Improvement) If you want to succeed, you must make improvement part of your plan.
Offer the lawn, plus a little more. Become better friends than you were with your last neighbors. Don't create a habit that will require more work from you. Don't offer to cook dinner or weed the garden. Just aim your heart to make it all around better, even better than before, some how, some way.
When you ask, maybe take a small gift with you too the door, nothing of great value beyond a "housewarming" gesture. Instead of a gift, you could just appear extra happy and uplifting every time you meet. The gesture will add the needed value to your kindness. The lawn is the "yes", the extra is the "and".
Mentally prepare yourself. If the answer is no, be ready to be thankful and happy and find some other way to make it better than before. Either way, anyway, make things "level up".
Employ this Interpersonal tool of "yes, and", then you will most likely get to mow the lawn.
Life always moves forward; so must we. If you only aim to go back to the same, good "mowing" relationship as before, you may lack the "happy energy" in how you come across in your request, you may even come across creepy or nervous and earn the opposite effect.
You want to improve the status quo because the past was better. But, to improve status quo without improving the past can't happen; Life won't allow it. To improve the status quo, we must also improve on the past.
Telling how things always were with the old neighbors would be great, but it actually isn't necessary. If you set your heart to "just be friendly, even friendlier than before", you won't need any reason, not even the past as your reason. Your happiness will speak for itself. With the extra boost that comes with aiming higher every day, talk of the happy, old days will be a plus. So the past isn't taboo at all, just optional.
Don't just aim for a "yes"; aim for a "yes, and" by giving the "yes, and" yourself.
Agreement in improv kevinmullaney.com
Simply put, an improviser must agree to all facts and circumstances that their scene partner establishes via dialogue, behavior or action. If I say that I’m a plumber, you must agree that I’m a plumber. If you act like you are in car, I must accept that. If I say that we are in an airport bar, set down your luggage and grab a drink.
...he even talks about "Yes Anding"...
Beyond simply agreeing with their partner, players should add information to the scene with each action or line of dialogue, at least at the beginning. This is often called “yes-anding” your scene partner.