This is a good question, and I don't think it counts as a duplicate of the question about defining zero-arity methods with or without parentheses.
The difference between toInt and toLowerCase here is that toInt is a syntactic enrichment provided by the standard library's StringOps class. You can check this by using reify and showCode in the REPL:
scala> import scala.reflect.runtime.universe.{ reify, showCode }
import scala.reflect.runtime.universe.{reify, showCode}
scala> showCode(reify("1".toInt).tree)
res0: String = Predef.augmentString("1").toInt
This just desugars the enrichment method call and shows you what implicit conversion has been applied to support it.
If we look at the toInt method on StringOps, we see that it's defined without parentheses. As the answer to the possible duplicate question points out, if a zero-arity method in Scala is defined without parentheses, it can't be called with parentheses (although if it's defined with parentheses it can be called either way).
So that's why "1".toInt() doesn't work. If String were a normal Scala class following normal Scala naming conventions, "ABC".toLowerCase() wouldn't work either, since the method would be defined without parentheses, since it's not side-effecting (this is just a convention, but it tends to be pretty consistently applied in Scala code).
The problem is that String isn't actually a class in the Scala standard library—it's just java.lang.String. At the JVM level, there's no difference between a zero-arity method defined with parentheses and one defined without parentheses—this is purely a Scala distinction, and it's not encoded in the JVM method signature, but in a separate batch of metadata stored in the class file.
The Scala language designers decided to treat all zero-arity methods that aren't defined in Scala (and therefore don't have this extra metadata) as if they were defined with parentheses. Since the toLowerCase method on String is really and truly a method of the java.lang.String class (not a syntactic enrichment method like toInt), it falls into this category, which means you can call it either way.