Hammerbarn is a reference to a warehouse chain of hardware stores called Bunnings that have a red hammer as a logo.
Mum's (Chilli's) comment that Lucky's Dad's grass was looking greener might be more than metaphor. Lucky's Dad recently got a pizza oven at Hammerbarn. According to the Hanmerbarn employee, the pizza ovens can be found in "Aisle 300, left at the fake grass, if you hit the flamingos you've gone too far." It's not inconceivable that Lucky's Dad got fake grass while buying the pizza oven.
Opening title image: two yard gnome husbands (Hecuba and Gerald).
The quote from this episode, "if you've hit the flamingoes you've gone too far," is a subtle callback to the season 1 episode, The Sleepover. In that episode, Muffin (incredibly tired from missing a nap) steals pretend chips and drives a car into a lawn flamingo (thus hitting the flamingo and going too far).
Closing title animation: Winton's Dad eating sausages.