Bartoli, a former Wimbledon finalist, lost to Camila Giorgi 6-3, 6-2, and Paszek fell to French wild-card entry Virgine Razzano 6-1, 6-4.
Bartoli dropped her serve six times in the match against the 88th-ranked Italian, who had four aces.
Also, American Christina McHale lost to Lauren Davis 7-5, 6-3.
Third-seeded Alize Cornet converted all four breakpoints to win 6-3, 6-2 against Mathilde Johansson, No. 5 Monica Niculescu of Romania rallied past Marina Erakovic 6-7 (7), 6-3, 7-5 (5) and No. 7 Chanelle Scheepers downed Claire Feuerstein 7-5, 6-4.
Flavia Pannetta, Eugenie Bouchard, Johanna Larsson, Lucie Hradecka, Magda Linette and Shelby Rogers advanced.
Heavy rain prevented play on Monday, forcing seven first-round matches to be moved to Tuesday.