The pair have also become closer on a personal level, with Florence sharing the intimate details of her late fiancé's death over a romantic plate of chicken and chips.
DI Parker, meanwhile, has also opened up about his father's passing, as well as some of his past dating nightmares.
The preview for next week's episode, however, now has fans absolutely convinced that the two characters are about to become romantically involved.
BBC
"Florence looks like she's on a date?" tweeted one person, noting that she appeared to be dressed for the occasion in the preview.
Another wrote: "The Neville & Florence romance storyline starts next week I'm guessing?"