where is nearby parking

Crossing the street behind a 7eleven

Street parking.

You can valet at Seattle Center

i will suggest you go a bit earlier (before 10am) or later (after 8pm), there should be plenty of street parking around (depend on the hours/days, it can be FOC).

Theres also street parking across from the MoPoP museum if you don't mind walking a block or 2. Usually its about $1 an hour a max of 10 hours.

Nearby within the compound

The tower is near the center in lower queen ane to ten minutes there is paid parking

Plenty all around. Just be prepared to pay for it. Most lots charge $9 an hour and you can pay by phone or credt/debit machines. Parking at the Needle is valet only and a bit more expensive.