In the context of the Internet and computing, parallel means more than one event happening at a time. It is usually contrasted with serial, meaning only one event happening at a time. In data transmission, the techniques of time division and space division are used, where time separates the transmission of individual bits of information sent serially and space (in multiple lines or paths) can be used to have multiple bits sent in parallel.
In the context of computer hardware and data transmission, serial connection, operation, and media usually indicate a simpler, slower operation (think of your serial mouse attachment). Parallel connection and operation (think of multiple characters being sent to your printer) indicates faster operation. This indication doesn't always hold since a serial medium (for example, fiber optic cable) can be much faster than a slower medium that carries multiple signals in parallel.
A conventional phone connection is generally thought of as a serial line since its usual transmission protocol is serial.
Conventional computers and their programs operate in a serial manner, with the computer reading a program and performing its instructions one after the other. However, some of today's computers have multiple processors that divide up the instructions and perform them in parallel.