January?... We get hit with snow especially hard in February, lol. Hmm, Westlake is next to me - I may attend. Since you will be staying in the Westlake area, you may consider some west-side sites.

Huntington Beach is in Bay Village (very close to Westlake). Not as "interesting" as Cleveland beaches, but perhaps more suburban and picturesque. Lake Erie is also cleaner the further you are from Cleveland proper.

The Rocky River Reservation in Cuyahoga County may also be interesting. Not as large as the Cuyahoga, but closer to Westlake. You can drive the parkway from Lakewood to North Royalton (and further, though it stops being a parkway at one point).

Westlake borders Lorain County, where some good "picturesque" parks can be found in Elyria, probably 20-30 minutes away (though areas of Lorain County do get more rural and pretty, you would be traveling even further away). They have the Black River with a number of parks, but no connecting parkway. The more interesting areas in those parks require some walking off the beaten-path (or should I say, official path), so it might not be easy if there is snow.