Parking in Cologne near train station/drive down the Rhine

"We are staying near the main train station in Cologne for several days in April."

The most scenic and interesting part of the Rhine Valley is the Upper Middle Rhine Valley south of Koblenz.

If you have the chance to stay THERE for a couple of those nights instead of all those nights in Cologne, I'd strongly urge you to do so. This is one of Germany's very best places to visit - with too much to pack into a single day. Cologne is alright but pales in comparison. A few images:

https://www.youtube.com/watch?v=U06AcB6a2Eg
https://i2.wp.com/herbert-piel.de/wp-44b92-content/uploads/2016/01/Loreleytal.jpg?fit=960%2C641
https://www.weinhaus-weiler.de/wp-content/uploads/2016/08/DSC_2434_1920.jpg
https://www.youtube.com/watch?x-yt-cl=85114404&x-yt-ts=1422579428&v=qxFF80wORNQ#t=32
http://ext.pimg.tw/samlucky5711/1379089584-1923285561.jpg?v=1379089585
https://www.youtube.com/watch?v=J2IjTma6VRY
https://www.marksburg.de/en/circuit/

Although I agree with Russ 100%, I wouldn't worry about overnight parking. Most German cities have automated occupancy data from all their civic garages (often operated by contractors, not the government) displayed at highway ramps and around towns on glowing signboards. I suggest that you rent the smallest car that will suit your needs, because parking stalls and garage aisles are narrower in Europe than they are in the U.S. Parking garage costs may be near U.S. levels. Forget about parking on the street in busy Cologne. I love Cologne and am not suggesting that you not spend a few days there!

Keep in mind that Germany has a superb train system, from which you will have a better view than from highways. It's a full half-day drive to Hanover from Cologne. Unless you have a specific reason for going that far (my own great-grandfather came from there), I'd consider Dusseldorf, Munster, and Kassel as big cities that are closer. If you decide on the recommended Middle Rhine, there are lots of attractive small cities near Frankfurt. It's the other way, but if I had a car in Cologne, I'd want to go to Monschau, which pairs better with Belgium or the Netherlands. The car is a hindrance in any large city.

In April, the Rhine day-boats may not have begun service yet (check KD website), but the view from the river is far superior than either from the roads or the trains.

Rather than renting a car, you can buy a Quer durchs Land Ticket for 52 EUR. The KD boats will be running on Monday April 15.

Take the 9:32 RE to Koblenz, then change to the 11:04 RE to Bacharach, getting you there by 11:35. Walk around town and have a quick lunch, then take the 1:06 pm RB to Bingen Stadt, getting you there by 1:26 pm. Walk around a bit and then down to the dock and get the 2:30 pm KD boat to Boppard. Show your train ticket and get a discount for the ride. You'll see all the best parts of the Rhine, arriving at Boppard at 4:50 pm. Explore the town, maybe have dinner, then get a train back to Cologne. There are at least 2 trains per hour to get you back until 9:12 pm, then every hour until 10:51 pm.

The QdL ticket allows you to hop on and off regional trains. And since you are not driving, you can enjoy the Rhine wine along the way.

That is just one off-the-top-of-my-head itinerary. Russ can probably come up with some others, especially if you want to visit one of the castles.

Russ, thank you for the links - they're fun, and very helpful. I'm sure it would be wonderful to stay in one of the towns, but we already have paid for a place in Cologne and we have limited time, so it will be just a day trip. You have helped us figure out how to make the best use of the time we do have.

Tim, that's good news about the parking, and yes, we will rent a small car. We're used to tight spaces in Seattle but we will expect even tighter. To explain what we're doing, the basic itinerary for the trip is 5 days in Leiden after landing at Schilpohl, then train to Ghent for another 5 days, then train to Cologne for 3 days. Then rent a car and drive to Hannover for 5 days, return the car there and train to Amsterdam for a few days before flying home. In Leiden & Ghent we'll do day trips by train (Bruges, Amsterdam etc.)
The reason we're going to Hannover is because both my grandparents were born in villages near there. My parents went there in 1985. I've contacted one relative who speaks English, so I will see the house where my grandmother was born, etc. I realize Hanover isn't the most exciting town but I also have a friend there to visit. It's helpful to know we should allow a half day for the drive to Hannover because I have another friend I may try to see on the way there.
While in Cologne we may squeeze in a train trip to Dusseldorf or another city. For the Middle Rhine, perhaps we can drive to one of the small towns (like Boppard), park, get on a boat for 1 - 2 hours, then take a train back to the car. We will research it some more, and thanks for the note about boat service. It looks like there's a high and low season; we're at the tail end of the low season but there are still quite a few boats scheduled. Having used a car for work in Seattle and having lived in NYC for many years before that, I understand how difficult a car can be in a big city - that's why we won't rent one until we're halfway through the trip. I hope this all makes sense now!

Sam, thanks for the suggestions. I didn't explain, but we are renting a car for other reasons - see the reply to Tim above. Also as I said above, we might try driving down to a small town like Boppard, taking a boat south for an hour or two, then getting off and taking a train back to where we left the car. One of us doesn't drink so although I know the region has great wines, that's not a draw for us. I will research this some more - I know it's a lot to do in one day, but hopefully, one way or another, we can spend a pleasant day seeing some of what the Middle Rhine has to offer.

Nothing wrong with renting a car to get to where you want to go. I do it all the time. Just pointing out alternatives.
1. Save a day of rental costs.
2. Save fuel costs.
3. Save parking fees, both in Cologne and in points along the the Rhine Gorge. Free parking is a rarity in the rest of the world.

Both people get to view the scenery, not just the passenger.

Just a technicality, but if you start in Cologne and drive to the middle Rhein (St. Goar, Bacharach), you are going UP the Rhine, not down. Yes, down (south) on a map, but up in elevation. The Rhein flow down from the middle Rhein through Cologne (Köln) to the North Sea.

Amplifying Lee's comment, we took the KD boat from Bingen to Boppard, which takes less time (with the current) but is plenty slow enough to enjoy the sights. The train back to the car is fast, but you'll miss the small, boat discount, for presenting a rail ticket. I wonder if you can show the train ticket before you use it? Boppard is worth some time, too. You may be too late to get a big discount for advance purchase of Gent-Cologne. (Change in Brussels)

On your next trip, price an open-jaw air ticket. You are covering a lot of ground miles between sleeps.