Bits per second is the rate of transfer of information bits. The ratio of BPS to baud depends on the information coding scheme that you are using. For example, each character in asynchronous RS-232 coding includes a start and stop bit that are not counted as information bits, so the BPS rate is actually less than the baud rate. Present-day modems, on the other hand, use a set of discrete amplitude and phase values to encode multiple bits with each signal change. Parallel transfer is the simultaneous transmission of the eight bit-voltages that constitute a byte.
Serial transfer is the sequential transmission of the eight bit-voltages that constitute a byte.
In an asynchronous connection, information is transferred one character at a time with no regard for the timing between characters. The problem that arises is that every single transmitted character carries baggage with it, in the form of start bits, parity bits, and stop bits. In a synchronous connection, timing is everything. One of the lines in the serial connection is used as a clock, eliminating start bits and stop bits. Because I've never messed with synchronous connections, the material presented here covers only asynchronous stuff.