If your parents wouldn't mind spending about $7-$9 on parking, have them take you to the Spectator Deck. It's a great place for viewing the planes. If Photopoint didn't get so cheap, I would post some pics that I took from the deck. It's at the very end of Concourse B, and you can take scanners, cameras, and even tripods, and security won't question you!
As far as busiest times, in personal experiences, there's 3 main time periods.
First is morning, from 8 till about 11. There are a lot of interesting aircraft at this time period, such as 727's, Dash-8's, 757's, and BAe-146's, as well as the familiar 737's and ERJ's.
From 11 until about 1, there is a lull in operations. At 1 until 4, is the second busy period. This is mainly an onslaught of 737's almost all from Continental. On a clear day, you may be able to see 4 or 5 737's lined up to land. There are also lonh lines to take off, too. Seeing 8 737's mixed with the ERJ's holding for takeoff is not uncommon during this time period. You may also see an occaisional stranger make a surprise visit at this time, but don't count on it. Two strangers I saw were a C-130 Hercules and a Douglas DC-3!
From 4 until about 7 is another lull. From 7 to 9, is the final busy period. Another sworm of odd planes arrive at CLE during this time. Southwest has a lot of flights into CLE during this time, as well as Midwest Express, and the charter airlines, Sunworld and ATA. Last Thursday, I was at the mound over by the 100th Bomb Group Restaurant, from 7 to 9, and I saw a lot of planes. Most planes were landing. I counted 6 lines up to land! Included in these was a Skyways Do-328JET! Also, this is the time when fully loaded MD-80's take off on the shorter runway and take-off around the touchdown point on the opposite end of the runway! Talk about deafining! Having an MD-80 pass 100 feet over your head is just amazing!!
As a side note, the only widebody aircraft you will see are FedEx's 2 A300's and UPS's A300. These are not moved during the day. They are loaded in the evening and they take-off at night. The FedEx planes can be seen from Cargo Rd., and the UPS planes (A300 and DC-8) can be seen from W. Hangar Rd. There currently is no scheduled widebody service to CLE, although rumor has it that United has 767-200's on their 6:30 AM CLE-ORD flight and on their 11:30 PM ORD-CLE flight. Another rumor is that Continental will change the LGW-CLE and CLE-LGW flights to 767-200's at the end of the year. The only reason I can figure that the airlines don't have widebodies into CLE, is because the plane would have to stop at a larger airport to receive all fuel needed, since CLE's longest runway is 9000 ft. The expansion project is under way, I will post about that at another time. The largest plane's scheduled to CLE are UA's and CO's 757-200's, and CO's 737-800's.
I hope this helps with your question!