you're getting it, but you need some minor clarifications. When you say the control signals are in parallel with the data signals, which are transmitted serially, I think you get it... but you are just wording it awkwardly. The control signals originate from the UART (at either end) and are converted to the RS232 line voltages (logic 0: -3 to -15V, logic 1: +3 to +15V) by a level convertor device like Hideki mentions, the MAX232. The TXD and RXD also originate from the UART, at each end, and are also converted to the RS232 line voltages by the MAX232 chip, where (logic 0: +3 to +15V, logic 1: -3 to -15V) <<-- inverted.
all these lines are on the RS232 connector "in parallel" as you say, but you should not think of it that way. All these lines just form the complete set of required data and control lines for an RS232 connection. And there might be 8 wires, or 3 wires or 25 wires... depending on the complexity of the implementation, but we don't need to get into that. The reason we don't need to get into that is because it is indeed an aging standard, as Hideko says, and the control signals can almost always be ignored with todays high speed, deep memory, fast processor computers. Those control signals that we can ignore are useful for handshaking, when the receiving ends are too slow. That almost never happens these days, and most people who use RS232 nowadays take a shortcut and ignore the control lines, and only use a 3-wire connection of TXD, RXD and GND.
Now, that said... the TXD and RXD signal lines are bit asynchronous, meaning that they include the start bit, data bits, any parity and the stop bits in one frame, that is asynchronous (out of sync) with any other frame. The timing of one entire frame, and one frame only, consisting of [start/data/parity/stop] is defined by the transition of the start bit from MARK to SPACE, and all these [start/data/parity/stop] bits are sent sequentially on the TX data line by the UART (and received on the RXD line by the other UART). It does not come from anywhere "deeper" as you wrote, the UART sends it. However, and this is where I think you need some clarification, the data bits that are in between those start and stop bits comes "from deeper" for example from your terminal program and from the CPU. Those data bits (which are usually 8-bit ASCII, without any start/stop information) come from the terminal program, via the CPU , and arrive at the UART. The UART then dutifully frames them with a start and stop bit and sequentially sends it out the TX data line. It does the same with the next 8-bit data byte that arrives at the UART "from deeper" within the computer. I am using your wording here, because I think that's how you understand it, but just know that the UART is a parallel to serial convertor at the transmitting end, and a serial to parallel converter at the receiving end. The serial conversion is done with start/stop framing bits, and the receiving end looks for those extra start/stop bits and removes them. So an 8-bit byte sent from one side ends up across the communications link and is recovered and turned back into an 8-bit parallel byte at the receiving end.
And remember, the communications link is not part of the UART, or part of ASCII-- it is just the medium through which the bits travel. There are many different communications links, and many different ways that the actual meaningful data bytes are represented on that link. RS232 uses asynchronous framing with start and stop bits around each data byte. Other communications links will use other, often more complex, methods.