Yes, you can stop and come back later. How the game saves depends on the game:

• Unsolved Case and Unboxing the Cryptic Killer save your progress between levels. If you stop in the middle of a level, you'll start that level again from the beginning.
• Parallel Experiment saves automatically all the time, and supports Cloud Saves on all platforms: your progress is backed up online, so you can continue on another device or recover your save if needed.

The save is stored only on the host's device, the player who selects Create Game. To continue later, the same player has to create the game again, and the partner joins with the new invitation code. The partner can be someone else this time. See How to start a game with your partner.

If one of you loses the connection, the game tries to reconnect automatically. If it can't, you'll return to the main menu. Create the game again and continue from the last save.

Lost your progress in Parallel Experiment? See Where to find Parallel Experiment save files.