I'm not sure if the current rating applies when you are wiring it as a current source rather than voltage regulator... The current rating is load max when operating as a voltage regulator. In the spec sheet for LM317, a circuit for a 1A current source is given, but I have no idea what kind of load is on the regulator when operating as a current source. It could even be that in that configuration it can carry more current, even..
If they can be used in parallel (Kirchoffs apply here?), then I'd just run three and call it good... I believe it is possible to run them side by side as voltage regulators, but in current regulation 'mode' I'll either need to research (or just try it).