I can echo what everyone has said so far. I personally have come to LOVE being at the park alone because I can do what I want when I want.
I have a lot of regret because I spent fifteen years barely visiting parks or riding coasters because I felt weird about going alone. After the first couple times, I got over the weirdness and now I sometimes get a little edgy if I've spent too much time at the park with friends because I miss the freedom of not always negotiating what to do next.
Going to the park alone doesn't have to be lonely, either. I've frequently gotten into conversations in lines or on the rare occasions I get paired with another rider, sometimes hitting it off with someone enough to ride two or three different rides in a row together. Being at the park as often as I am, I've made friends with a few other frequent single-riders, who I ride one or two rides and chat with before we go on our separate ways at the park. I'm not the most friendly person ordinarily, but enjoying myself, doing something I love in a place I love, tends to open me up more and put me in a more friendly mood. But on days I don't feel like talking to anybody, I can do that too.
It helps to have a prop to keep you busy especially until you feel more comfortable: a smartphone is good. It gives you something to do in line, and makes you feel a little more connected to the friends who aren't currently with you. I've become less dependent on mine than I was at the beginning, though, often choosing to people-watch and just look around at the details of the park instead.
Really, the thing I find most uncomfortable doing at the park alone is eating. So, I usually take my food out to the car (or brought snacks to eat in the car before I bought season-long dining) and talk to my husband on the phone while I eat.
Because I try to go to the park when it's not busy, it's uncommon for me to be paired up with someone by the line-separator ride-op. But when it happens, it's not really as weird as I worried it would be. If not told by an employee to pair up, I will not "hop on" with another solo rider unless I ask first: I just wait for the next train. This seems to be pretty much accepted behavior, and it is exceedingly rare someone just sits next to me without being directed by an employee or asking me if I mind. The exception to this is on Diamondback, in the even rows where the seats are widely spaced, I neither ask nor expect to be asked if it is ok. It's perfectly polite to join an even row with another single without saying anything because you're actually riding closer to the people in the odd row in front of you than you will be to the other person in your row.
My personal policy is to never refuse a request to sit with me if asked, even though I don't ask others except sometimes when I'm waiting for the front row. I would rather just wait a train than potentially make someone else feel awkward.