No, not really. My parents encouraged me to learn lots about computers - weren’t wild about me playing games on it when I had schoolwork, and at one point actually got a basic brother word processor so that I could type up school papers without getting ‘sidetracked’, but that’s more than fair in retrospect.
Then again, my father was also a computer professional - he was an associate professor of computer science at the local university. 
I first got my hands on a computer in the early 1970s at school, using an Olivetti Programma 101, and later took a course at school (O-level Computer Science) which gave me access to a remote mainframe with a BASIC interpreter. Teletype, acoustic coupler modem, plenty of paper tape and printouts. Magic stuff. Home computers were still not practical propositions.
I decided that (a) I liked doing this; (b) I was damned good at it, and (c) this was a career path that was going to be big, and that programmers would be in short supply. My parents supported me fully, and were very proud and encouraging.
By the time I was 18, I was spending a lot of time with my father, who was a teacher, writing educational and administrative software for his school, which had an Apple II with all the available add-ons. In those days, there was pretty much zero off-the-shelf software around that they could use.
I went on to have a successful career in IT, as an analyst/designer/programmer type of person. I have my parents to thank for their support.
I come from a very geek family. How geek, you ask? My parents met in the late '70s when they were both working for BBN Laboratories. My maternal grandfather was an architect for roughly eleven million years before he retired; my maternal grandmother was a supervisor in the punch card department at Harvard, back in the Cretaceous period when the sacred cards were looked after by a specially-trained secretarial pool. It was not until I got to college and ran into The Jargon File that I realized that a lot of sayings and expressions that were in everyday use in our house were, in fact, highly dorktastic and specific to the tech industry. I have geek street cred.
We were never really limited on computer time. I don’t think it crossed anyone’s mind, frankly. For starters, I can’t remember a time when there was only one computer in the house; the fanciest one was always Dad’s, because he did (and still does) CAD work at home, but even pre-internet there was always a second computer around with some primitive word processor on it, and a switch box for the printer so we could print homework if we wanted to, even while Dad was busy. Secondarily, nobody in the family is a big PC gamer, so the only time we ever had an issue with hogging a particular machine was when we got Myst (for Windows 95!) and I think all four of us had games in progress for a while.
Things did get a little touchy when we started getting BBSes and AOL/Internet access, but only because there were two modems and only one spare phone line. There were only two rules: Don’t incur any extra charges, and don’t tell strangers where you live. My parents really couldn’t have cared less if we were looking up porn or how to build bombs. They were probably happier about us getting it online than getting it from other people our age.
The actual squabbles took place over the Nintendo/Super Nintendo. The only TV in the house that was large enough to comfortably play games on was in the living room, which usually meant that we kids got booted on Sunday nights so my parents didn’t miss Hercule Poirot on PBS. This was also occasionally a four-way fight on other nights of the week; Mom and Dad actually bought the NES for themselves, because they wanted to play Zelda when it came out, and getting Dad to let go of the controller was an uphill battle. (Getting Dad to let go of the controller is still an uphill battle. I gave my parents DragonQuest VIII for Christmas last year and they put 24 hours on the game clock in two days. I didn’t hear word one from them for about a week.)
My BA isn’t in anything that most people would even remotely connect to electronics, but being as sociologists aren’t in great demand right now, my most marketable job skill is the ability to perform computer voodoo. The family considers my sister to be ‘not terribly interested’ in computers, but by the family definition ‘not terribly interested’ still means ‘able to bend MS Office to her will, and replace her own hard drive without help’.
There was no real need to limit our access to Pong.
I was encouraged to learn how things worked by taking them apart and putting them back together again, and I had a huge kick-ass chemistry set, but managed to end up in CSci instead of mechanical or chemical engineering.