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Parity Party XI

Normal Sudoku rules apply. No digit may repeat in a row, column, or box.

Also, the grey cells are 'fortress' cells; these all contain values that are larger than what is in the orthogonally neighboring white cells.

Special rules of Parity Party: A number at the edge of the grid indicates the sum of cells encountered when moving through the grid from that vantage point, UP TO AND INCLUDING the first odd or first even digit. For example, a 15 at the bottom of a column could indicate something like 7-8, 8-7, 1-3-5-6 or 2-6-4-3 (among others) when reading up that column... but not 1-2-3-4-5 or 2-6-3-4 etc. That is, the parity of the farthest digit must deviate from that of the others.

Note: If the clue is less than 10, it might simply indicate the value of the first cell. That is to say, you could be adding up zero odd and one even number, or vice versa.