#$$$$$----------------------------------------------------
# convert everything to string
overlap <- "> 2"
pre <- "as.numeric(rowSums(DF[, 1:6]) "
post <- ")"

# paste together, then wrap in expression
expr <- as.expression(parse(text=paste0(pre, overlap, post)))

# call eval
eval(expr)
#$$$$$----------------------------------------------------



You have to convert the entire expression to a string, then convert the parsed text to an expression. 
Finally, call eval() on the expression. 

eg: 

    overlap <- "> 2"

    # Convert to string
    exprAsString <- paste0("as.numeric(rowSums(DF[, 1:6]) ", overlap, ")")

    # Convert to expression, using parse
    expr <- as.expression(parse(text=exprAsString))

    # Then call eval()
    eval(expr)


## CONFIRMING IT WORKS:

    identical(eval(expr), as.numeric(rowSums(DF[, 1:6]) > 2))
    # [1] TRUE