Since both were promoted back to Serie A last season, they have dropped the most points from winning positions - no fewer than 23 combined - leaving them stuck near the foot of the standings.
Match preview
Days after staging an incredible fightback to record just their second win of the season, Venezia travelled to San Siro, where they were faced by reigning champions Inter Milan last weekend.
The Lagunari had just scrapped their way to a 3-2 home win over Udinese after trailing by two goals, but they were put through a tougher test by the Scudetto holders.
Trailing 1-0 deep into second-half stoppage time, Marin Sverko seemed to have bundled in a last-gasp leveller, but the VAR then chalked it off for handball and the final whistle immediately followed.
Last season's Serie B playoff winners have still won just twice across all competitions since Eusebio Di Francesco took charge in the summer, and they are languishing inside the drop zone with more than a quarter of the season played.
Venezia have scored 10 goals in 11 league matches, though, and that represents the club's best tally at this stage of a top-tier campaign since 1962 - albeit they have mostly dwelled in the lower divisions during that time.
As the Arancioneroverdi now attempt to win back-to-back Serie A home matches for the first time in nearly 25 years, they will meet opponents who have yet to taste top-flight success in Venice.
Parma were also beaten 3-2 by Venezia in last term's Serie B meeting between the teams at Stadio Penzo, so, in several senses, they will seek an overdue victory on Saturday.
On Monday night, the Gialloblu failed to pull four points clear of their upcoming hosts, as they were beaten 1-0 by Genoa on home turf, leaving them just outside the relegation zone.
One of seven teams split by one point at the wrong end of the table, Fabio Pecchia's side last won in August, having effectively let 12 points slip from their grasp so far.
As a result, last season's second-tier champions have drawn each of their last four away fixtures and could now record five straight Serie A stalemates for the very first time.
However, Pecchia is intent on picking up maximum points at the Penzo: going toe to toe with a direct relegation rival will raise the stakes, meaning victory would taste even sweeter.
Venezia Serie A form:
L
L
L
D
W
L
Parma Serie A form:
L
D
D
D
D
L
Team News
After making a handful of changes against Inter, Eusebio Di Francesco is expected to recall Gianluca Busio on Saturday, with the USA international set to join Gaetano Oristanio in support of Joel Pohjanpalo.
The Finnish striker leads Venezia's front line, having scored three of his four league goals at Stadio Penzo this season; over the last three seasons, no player has netted more goals on home soil (28) between Italy's top two divisions.
Once again, Bjarki Bjarkason is the Lagunari's sole absentee, but Parma will have several men missing.
Hernani sustained a muscular injury last week, while fellow midfielder Adrian Bernabe injured his thigh flexor against Genoa and could be ruled out for up to a month.
The latter's replacement on Monday evening, teenager Anas Haj Mohamed, subsequently sprained his ankle, so they all join ACL victim Alessandro Circati on the sidelines.
In attack, Ange-Yoan Bonny should link up with Dennis Man, who has been directly involved in five league goals this season - three of his own and two assists - making him Parma's most productive player.
Venezia possible starting lineup:
Stankovic; Candela, Idzes, Svoboda; Zampano, Duncan, Nicolussi Caviglia, Haps; Busio, Oristanio; Pohjanpalo
Parma possible starting lineup:
Suzuki; Coulibaly, Del Prato, Balogh, Valeri; Keita, Estevez; Man, Sohm, Mihaila; Bonny
We say: Venezia 2-2 Parma
Though Parma are more potent than their hosts up front, they have consistently paid for some costly slip-ups, losing seven points from leading positions away from home.