The 25-year-old inked a two-year extension in 2021 to take himself to free agency for this year, with Parish falling under the restricted band given his place among the top 25% of earners at Tullamarine.
With the AFL now reaching its bye rounds period, off-contract players across the competition are expected to commence contract discussions before the back end of the season commences.
According to reporter Tom Browne, Parish's future could be made clearer in the coming weeks, with signs pointing to a contract extension at Bomberland.
"Having said that, I spoke to some sources connected to discussions with the Bombers, and they think that Parish wants to stay at Essendon. He's got a partner in Melbourne, he's got family on the outskirts of Geelong.
"(Sources) think he wants to stay at the Bombers and there could be movement there, as far as Essendon's concerned, in the next couple of weeks.
"My take from all that, there's positive signs he wants to stay at Essendon. Geelong are naturally in the market, that's their modus operandi - bringing guys from Geelong back to the area.
"But there's certainly positive signs Parish will recommit to the Bombers. If he doesn't in the next week or so I think that would spell alarm bells."
— Triple M Footy (@triplemfooty) June 16, 2023
Parish isn't the only restricted free agent on Essendon's books, with defender Mason Redman also said to be garnering interest from South Australia.
Adelaide are seen as the leading rival suitor for the in-form Bomber, while Essendon are hopeful of retaining their fiery flanker.