It's possible...but you'd basically be building an IC out of discrete transistors which makes it completely impractical. I think the simplest one you could do would still require a 555 timer and that has flip flops. FLip flops themselves are integrated circuits so you'd have to build each one out of a bunch of discrete transistors, as well as a voltage comparator (which is similar to an op-amp). THese are also ICs and require many transistors. I think it works out to be a total of about 20 to 45 transistors for the 555 timer, depending on the comparator you choose to make.
Why can't you use an IC? If you're just looking for a challenge, building a simple op-amp out of regular transistors is hard enough.
What do you mean parallel circuit anyways? THat does't make any sense.